so(16), type \(D^{1}_8\)
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Page generated by the calculator project.
Lie algebra type: D^{1}_8.
Weyl group size: 5160960.
A drawing of the root system in its corresponding Coxeter plane. Computations were carried out as explained by John Stembridge.
The darker red dots can be dragged with the mouse to rotate the picture.
The grey lines are the edges of the Weyl chamber.
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The root system has 112 elements.
Simple basis coordinatesEpsilon coordinatesReflection w.r.t. root
(-1, -2, -2, -2, -2, -2, -1, -1)-e_{1}-e_{2}\(s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -2, -2, -2, -2, -1, -1)-e_{1}-e_{3}\(s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, -2, -2, -2, -2, -1, -1)-e_{2}-e_{3}\(s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(-1, -1, -1, -2, -2, -2, -1, -1)-e_{1}-e_{4}\(s_{1}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}s_{4}\)
(0, -1, -1, -2, -2, -2, -1, -1)-e_{2}-e_{4}\(s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}s_{5}s_{4}\)
(-1, -1, -1, -1, -2, -2, -1, -1)-e_{1}-e_{5}\(s_{1}s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}\)
(0, 0, -1, -2, -2, -2, -1, -1)-e_{3}-e_{4}\(s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}s_{5}s_{4}\)
(0, -1, -1, -1, -2, -2, -1, -1)-e_{2}-e_{5}\(s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}s_{5}\)
(-1, -1, -1, -1, -1, -2, -1, -1)-e_{1}-e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}\)
(0, 0, -1, -1, -2, -2, -1, -1)-e_{3}-e_{5}\(s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}s_{5}\)
(0, -1, -1, -1, -1, -2, -1, -1)-e_{2}-e_{6}\(s_{2}s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}\)
(-1, -1, -1, -1, -1, -1, -1, -1)-e_{1}-e_{7}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}\)
(0, 0, 0, -1, -2, -2, -1, -1)-e_{4}-e_{5}\(s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{7}s_{6}s_{5}\)
(0, 0, -1, -1, -1, -2, -1, -1)-e_{3}-e_{6}\(s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}\)
(0, -1, -1, -1, -1, -1, -1, -1)-e_{2}-e_{7}\(s_{2}s_{3}s_{4}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}\)
(-1, -1, -1, -1, -1, -1, 0, -1)-e_{1}-e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(-1, -1, -1, -1, -1, -1, -1, 0)-e_{1}+e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, 0, -1, -1, -2, -1, -1)-e_{4}-e_{6}\(s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{7}s_{6}\)
(0, 0, -1, -1, -1, -1, -1, -1)-e_{3}-e_{7}\(s_{3}s_{4}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{7}\)
(0, -1, -1, -1, -1, -1, 0, -1)-e_{2}-e_{8}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(0, -1, -1, -1, -1, -1, -1, 0)-e_{2}+e_{8}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -1, -1, -1, -1, 0, 0)-e_{1}+e_{7}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, -1, -2, -1, -1)-e_{5}-e_{6}\(s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{7}s_{6}\)
(0, 0, 0, -1, -1, -1, -1, -1)-e_{4}-e_{7}\(s_{4}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{7}\)
(0, 0, -1, -1, -1, -1, 0, -1)-e_{3}-e_{8}\(s_{3}s_{4}s_{5}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}\)
(0, 0, -1, -1, -1, -1, -1, 0)-e_{3}+e_{8}\(s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, -1, -1, -1, -1, 0, 0)-e_{2}+e_{7}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -1, -1, -1, 0, 0, 0)-e_{1}+e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, -1, -1, -1, -1)-e_{5}-e_{7}\(s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{7}\)
(0, 0, 0, -1, -1, -1, 0, -1)-e_{4}-e_{8}\(s_{4}s_{5}s_{6}s_{8}s_{6}s_{5}s_{4}\)
(0, 0, 0, -1, -1, -1, -1, 0)-e_{4}+e_{8}\(s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}\)
(0, 0, -1, -1, -1, -1, 0, 0)-e_{3}+e_{7}\(s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, -1, -1, -1, 0, 0, 0)-e_{2}+e_{6}\(s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -1, -1, 0, 0, 0, 0)-e_{1}+e_{5}\(s_{1}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, 0, -1, -1, -1)-e_{6}-e_{7}\(s_{7}s_{6}s_{8}s_{6}s_{7}\)
(0, 0, 0, 0, -1, -1, 0, -1)-e_{5}-e_{8}\(s_{5}s_{6}s_{8}s_{6}s_{5}\)
(0, 0, 0, 0, -1, -1, -1, 0)-e_{5}+e_{8}\(s_{5}s_{6}s_{7}s_{6}s_{5}\)
(0, 0, 0, -1, -1, -1, 0, 0)-e_{4}+e_{7}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(0, 0, -1, -1, -1, 0, 0, 0)-e_{3}+e_{6}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, -1, -1, -1, 0, 0, 0, 0)-e_{2}+e_{5}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(-1, -1, -1, 0, 0, 0, 0, 0)-e_{1}+e_{4}\(s_{1}s_{2}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, 0, -1, 0, -1)-e_{6}-e_{8}\(s_{6}s_{8}s_{6}\)
(0, 0, 0, 0, 0, -1, -1, 0)-e_{6}+e_{8}\(s_{6}s_{7}s_{6}\)
(0, 0, 0, 0, -1, -1, 0, 0)-e_{5}+e_{7}\(s_{5}s_{6}s_{5}\)
(0, 0, 0, -1, -1, 0, 0, 0)-e_{4}+e_{6}\(s_{4}s_{5}s_{4}\)
(0, 0, -1, -1, 0, 0, 0, 0)-e_{3}+e_{5}\(s_{3}s_{4}s_{3}\)
(0, -1, -1, 0, 0, 0, 0, 0)-e_{2}+e_{4}\(s_{2}s_{3}s_{2}\)
(-1, -1, 0, 0, 0, 0, 0, 0)-e_{1}+e_{3}\(s_{1}s_{2}s_{1}\)
(0, 0, 0, 0, 0, 0, 0, -1)-e_{7}-e_{8}\(s_{8}\)
(0, 0, 0, 0, 0, 0, -1, 0)-e_{7}+e_{8}\(s_{7}\)
(0, 0, 0, 0, 0, -1, 0, 0)-e_{6}+e_{7}\(s_{6}\)
(0, 0, 0, 0, -1, 0, 0, 0)-e_{5}+e_{6}\(s_{5}\)
(0, 0, 0, -1, 0, 0, 0, 0)-e_{4}+e_{5}\(s_{4}\)
(0, 0, -1, 0, 0, 0, 0, 0)-e_{3}+e_{4}\(s_{3}\)
(0, -1, 0, 0, 0, 0, 0, 0)-e_{2}+e_{3}\(s_{2}\)
(-1, 0, 0, 0, 0, 0, 0, 0)-e_{1}+e_{2}\(s_{1}\)
(1, 0, 0, 0, 0, 0, 0, 0)e_{1}-e_{2}\(s_{1}\)
(0, 1, 0, 0, 0, 0, 0, 0)e_{2}-e_{3}\(s_{2}\)
(0, 0, 1, 0, 0, 0, 0, 0)e_{3}-e_{4}\(s_{3}\)
(0, 0, 0, 1, 0, 0, 0, 0)e_{4}-e_{5}\(s_{4}\)
(0, 0, 0, 0, 1, 0, 0, 0)e_{5}-e_{6}\(s_{5}\)
(0, 0, 0, 0, 0, 1, 0, 0)e_{6}-e_{7}\(s_{6}\)
(0, 0, 0, 0, 0, 0, 1, 0)e_{7}-e_{8}\(s_{7}\)
(0, 0, 0, 0, 0, 0, 0, 1)e_{7}+e_{8}\(s_{8}\)
(1, 1, 0, 0, 0, 0, 0, 0)e_{1}-e_{3}\(s_{1}s_{2}s_{1}\)
(0, 1, 1, 0, 0, 0, 0, 0)e_{2}-e_{4}\(s_{2}s_{3}s_{2}\)
(0, 0, 1, 1, 0, 0, 0, 0)e_{3}-e_{5}\(s_{3}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 0, 0, 0)e_{4}-e_{6}\(s_{4}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 0, 0)e_{5}-e_{7}\(s_{5}s_{6}s_{5}\)
(0, 0, 0, 0, 0, 1, 1, 0)e_{6}-e_{8}\(s_{6}s_{7}s_{6}\)
(0, 0, 0, 0, 0, 1, 0, 1)e_{6}+e_{8}\(s_{6}s_{8}s_{6}\)
(1, 1, 1, 0, 0, 0, 0, 0)e_{1}-e_{4}\(s_{1}s_{2}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 0, 0, 0, 0)e_{2}-e_{5}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 0, 0, 0)e_{3}-e_{6}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1, 0, 0)e_{4}-e_{7}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 1, 0)e_{5}-e_{8}\(s_{5}s_{6}s_{7}s_{6}s_{5}\)
(0, 0, 0, 0, 1, 1, 0, 1)e_{5}+e_{8}\(s_{5}s_{6}s_{8}s_{6}s_{5}\)
(0, 0, 0, 0, 0, 1, 1, 1)e_{6}+e_{7}\(s_{7}s_{6}s_{8}s_{6}s_{7}\)
(1, 1, 1, 1, 0, 0, 0, 0)e_{1}-e_{5}\(s_{1}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 1, 0, 0, 0)e_{2}-e_{6}\(s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 1, 0, 0)e_{3}-e_{7}\(s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1, 1, 0)e_{4}-e_{8}\(s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}\)
(0, 0, 0, 1, 1, 1, 0, 1)e_{4}+e_{8}\(s_{4}s_{5}s_{6}s_{8}s_{6}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 1, 1)e_{5}+e_{7}\(s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{7}\)
(1, 1, 1, 1, 1, 0, 0, 0)e_{1}-e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 1, 1, 0, 0)e_{2}-e_{7}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 1, 1, 0)e_{3}-e_{8}\(s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(0, 0, 1, 1, 1, 1, 0, 1)e_{3}+e_{8}\(s_{3}s_{4}s_{5}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1, 1, 1)e_{4}+e_{7}\(s_{4}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{7}\)
(0, 0, 0, 0, 1, 2, 1, 1)e_{5}+e_{6}\(s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{7}s_{6}\)
(1, 1, 1, 1, 1, 1, 0, 0)e_{1}-e_{7}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 1, 1, 1, 0)e_{2}-e_{8}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(0, 1, 1, 1, 1, 1, 0, 1)e_{2}+e_{8}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 1, 1, 1)e_{3}+e_{7}\(s_{3}s_{4}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{7}\)
(0, 0, 0, 1, 1, 2, 1, 1)e_{4}+e_{6}\(s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{7}s_{6}\)
(1, 1, 1, 1, 1, 1, 1, 0)e_{1}-e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(1, 1, 1, 1, 1, 1, 0, 1)e_{1}+e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 1, 1, 1, 1)e_{2}+e_{7}\(s_{2}s_{3}s_{4}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}\)
(0, 0, 1, 1, 1, 2, 1, 1)e_{3}+e_{6}\(s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}\)
(0, 0, 0, 1, 2, 2, 1, 1)e_{4}+e_{5}\(s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{7}s_{6}s_{5}\)
(1, 1, 1, 1, 1, 1, 1, 1)e_{1}+e_{7}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}\)
(0, 1, 1, 1, 1, 2, 1, 1)e_{2}+e_{6}\(s_{2}s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}\)
(0, 0, 1, 1, 2, 2, 1, 1)e_{3}+e_{5}\(s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}s_{5}\)
(1, 1, 1, 1, 1, 2, 1, 1)e_{1}+e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}\)
(0, 1, 1, 1, 2, 2, 1, 1)e_{2}+e_{5}\(s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}s_{5}\)
(0, 0, 1, 2, 2, 2, 1, 1)e_{3}+e_{4}\(s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{7}s_{6}s_{5}s_{4}\)
(1, 1, 1, 1, 2, 2, 1, 1)e_{1}+e_{5}\(s_{1}s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}\)
(0, 1, 1, 2, 2, 2, 1, 1)e_{2}+e_{4}\(s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}s_{5}s_{4}\)
(1, 1, 1, 2, 2, 2, 1, 1)e_{1}+e_{4}\(s_{1}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}s_{4}\)
(0, 1, 2, 2, 2, 2, 1, 1)e_{2}+e_{3}\(s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(1, 1, 2, 2, 2, 2, 1, 1)e_{1}+e_{3}\(s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}s_{4}s_{3}\)
(1, 2, 2, 2, 2, 2, 1, 1)e_{1}+e_{2}\(s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{6}s_{8}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}\)
Comma delimited list of roots: (-1, -2, -2, -2, -2, -2, -1, -1), (-1, -1, -2, -2, -2, -2, -1, -1), (0, -1, -2, -2, -2, -2, -1, -1), (-1, -1, -1, -2, -2, -2, -1, -1), (0, -1, -1, -2, -2, -2, -1, -1), (-1, -1, -1, -1, -2, -2, -1, -1), (0, 0, -1, -2, -2, -2, -1, -1), (0, -1, -1, -1, -2, -2, -1, -1), (-1, -1, -1, -1, -1, -2, -1, -1), (0, 0, -1, -1, -2, -2, -1, -1), (0, -1, -1, -1, -1, -2, -1, -1), (-1, -1, -1, -1, -1, -1, -1, -1), (0, 0, 0, -1, -2, -2, -1, -1), (0, 0, -1, -1, -1, -2, -1, -1), (0, -1, -1, -1, -1, -1, -1, -1), (-1, -1, -1, -1, -1, -1, 0, -1), (-1, -1, -1, -1, -1, -1, -1, 0), (0, 0, 0, -1, -1, -2, -1, -1), (0, 0, -1, -1, -1, -1, -1, -1), (0, -1, -1, -1, -1, -1, 0, -1), (0, -1, -1, -1, -1, -1, -1, 0), (-1, -1, -1, -1, -1, -1, 0, 0), (0, 0, 0, 0, -1, -2, -1, -1), (0, 0, 0, -1, -1, -1, -1, -1), (0, 0, -1, -1, -1, -1, 0, -1), (0, 0, -1, -1, -1, -1, -1, 0), (0, -1, -1, -1, -1, -1, 0, 0), (-1, -1, -1, -1, -1, 0, 0, 0), (0, 0, 0, 0, -1, -1, -1, -1), (0, 0, 0, -1, -1, -1, 0, -1), (0, 0, 0, -1, -1, -1, -1, 0), (0, 0, -1, -1, -1, -1, 0, 0), (0, -1, -1, -1, -1, 0, 0, 0), (-1, -1, -1, -1, 0, 0, 0, 0), (0, 0, 0, 0, 0, -1, -1, -1), (0, 0, 0, 0, -1, -1, 0, -1), (0, 0, 0, 0, -1, -1, -1, 0), (0, 0, 0, -1, -1, -1, 0, 0), (0, 0, -1, -1, -1, 0, 0, 0), (0, -1, -1, -1, 0, 0, 0, 0), (-1, -1, -1, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, -1, 0, -1), (0, 0, 0, 0, 0, -1, -1, 0), (0, 0, 0, 0, -1, -1, 0, 0), (0, 0, 0, -1, -1, 0, 0, 0), (0, 0, -1, -1, 0, 0, 0, 0), (0, -1, -1, 0, 0, 0, 0, 0), (-1, -1, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, -1), (0, 0, 0, 0, 0, 0, -1, 0), (0, 0, 0, 0, 0, -1, 0, 0), (0, 0, 0, 0, -1, 0, 0, 0), (0, 0, 0, -1, 0, 0, 0, 0), (0, 0, -1, 0, 0, 0, 0, 0), (0, -1, 0, 0, 0, 0, 0, 0), (-1, 0, 0, 0, 0, 0, 0, 0), (1, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 1), (1, 1, 0, 0, 0, 0, 0, 0), (0, 1, 1, 0, 0, 0, 0, 0), (0, 0, 1, 1, 0, 0, 0, 0), (0, 0, 0, 1, 1, 0, 0, 0), (0, 0, 0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 0, 1, 1, 0), (0, 0, 0, 0, 0, 1, 0, 1), (1, 1, 1, 0, 0, 0, 0, 0), (0, 1, 1, 1, 0, 0, 0, 0), (0, 0, 1, 1, 1, 0, 0, 0), (0, 0, 0, 1, 1, 1, 0, 0), (0, 0, 0, 0, 1, 1, 1, 0), (0, 0, 0, 0, 1, 1, 0, 1), (0, 0, 0, 0, 0, 1, 1, 1), (1, 1, 1, 1, 0, 0, 0, 0), (0, 1, 1, 1, 1, 0, 0, 0), (0, 0, 1, 1, 1, 1, 0, 0), (0, 0, 0, 1, 1, 1, 1, 0), (0, 0, 0, 1, 1, 1, 0, 1), (0, 0, 0, 0, 1, 1, 1, 1), (1, 1, 1, 1, 1, 0, 0, 0), (0, 1, 1, 1, 1, 1, 0, 0), (0, 0, 1, 1, 1, 1, 1, 0), (0, 0, 1, 1, 1, 1, 0, 1), (0, 0, 0, 1, 1, 1, 1, 1), (0, 0, 0, 0, 1, 2, 1, 1), (1, 1, 1, 1, 1, 1, 0, 0), (0, 1, 1, 1, 1, 1, 1, 0), (0, 1, 1, 1, 1, 1, 0, 1), (0, 0, 1, 1, 1, 1, 1, 1), (0, 0, 0, 1, 1, 2, 1, 1), (1, 1, 1, 1, 1, 1, 1, 0), (1, 1, 1, 1, 1, 1, 0, 1), (0, 1, 1, 1, 1, 1, 1, 1), (0, 0, 1, 1, 1, 2, 1, 1), (0, 0, 0, 1, 2, 2, 1, 1), (1, 1, 1, 1, 1, 1, 1, 1), (0, 1, 1, 1, 1, 2, 1, 1), (0, 0, 1, 1, 2, 2, 1, 1), (1, 1, 1, 1, 1, 2, 1, 1), (0, 1, 1, 1, 2, 2, 1, 1), (0, 0, 1, 2, 2, 2, 1, 1), (1, 1, 1, 1, 2, 2, 1, 1), (0, 1, 1, 2, 2, 2, 1, 1), (1, 1, 1, 2, 2, 2, 1, 1), (0, 1, 2, 2, 2, 2, 1, 1), (1, 1, 2, 2, 2, 2, 1, 1), (1, 2, 2, 2, 2, 2, 1, 1) The resulting Lie bracket pairing table follows.
Type D^{1}_8.The letter \(\displaystyle h\) stands for elements of the Cartan subalgebra,
the letter \(\displaystyle g\) stands for the Chevalley (root space) generators of non-zero weight.
The generator \(\displaystyle h_i\) is the element of the Cartan subalgebra dual to the
i^th simple root, that is, \(\displaystyle [h_i, g] =\langle \alpha_i , \gamma\rangle g\),
where g is a Chevalley generator, \(\displaystyle \gamma\) is its weight, and
\(\displaystyle \alpha_i\) is the i^th simple root.
The Lie bracket table is too large to be rendered in LaTeX, displaying in html format instead.
roots simple coords epsilon coordinates[,]g_{-56}g_{-55}g_{-54}g_{-53}g_{-52}g_{-51}g_{-50}g_{-49}g_{-48}g_{-47}g_{-46}g_{-45}g_{-44}g_{-43}g_{-42}g_{-41}g_{-40}g_{-39}g_{-38}g_{-37}g_{-36}g_{-35}g_{-34}g_{-33}g_{-32}g_{-31}g_{-30}g_{-29}g_{-28}g_{-27}g_{-26}g_{-25}g_{-24}g_{-23}g_{-22}g_{-21}g_{-20}g_{-19}g_{-18}g_{-17}g_{-16}g_{-15}g_{-14}g_{-13}g_{-12}g_{-11}g_{-10}g_{-9}g_{-8}g_{-7}g_{-6}g_{-5}g_{-4}g_{-3}g_{-2}g_{-1}h_{1}h_{2}h_{3}h_{4}h_{5}h_{6}h_{7}h_{8}g_{1}g_{2}g_{3}g_{4}g_{5}g_{6}g_{7}g_{8}g_{9}g_{10}g_{11}g_{12}g_{13}g_{14}g_{15}g_{16}g_{17}g_{18}g_{19}g_{20}g_{21}g_{22}g_{23}g_{24}g_{25}g_{26}g_{27}g_{28}g_{29}g_{30}g_{31}g_{32}g_{33}g_{34}g_{35}g_{36}g_{37}g_{38}g_{39}g_{40}g_{41}g_{42}g_{43}g_{44}g_{45}g_{46}g_{47}g_{48}g_{49}g_{50}g_{51}g_{52}g_{53}g_{54}g_{55}g_{56}
(-1, -2, -2, -2, -2, -2, -1, -1)-e_{1}-e_{2}g_{-56}000000000000000000000000000000000000000000000000000000000g_{-56}0000000g_{-55}000000-g_{-54}g_{-53}00000-g_{-52}g_{-51}00000-g_{-49}g_{-48}0000-g_{-46}g_{-45}0000-g_{-42}g_{-41}g_{-40}00-g_{-37}-g_{-36}g_{-35}00-g_{-30}g_{-29}0-g_{-24}g_{-23}0-g_{-17}g_{-16}-g_{-10}g_{-9}-g_{-2}-h_{8}-h_{7}-2h_{6}-2h_{5}-2h_{4}-2h_{3}-2h_{2}-h_{1}
(-1, -1, -2, -2, -2, -2, -1, -1)-e_{1}-e_{3}g_{-55}000000000000000000000000000000000000000000000000000000g_{-56}0g_{-55}-g_{-55}g_{-55}00000g_{-54}0g_{-53}0000000g_{-51}0000-g_{-50}0g_{-48}0000-g_{-47}0g_{-45}000-g_{-43}0g_{-41}g_{-40}00-g_{-38}00g_{-35}0-g_{-32}-g_{-31}0g_{-29}0-g_{-25}0g_{-23}-g_{-18}0g_{-16}-g_{-11}0-g_{-3}-g_{-1}-h_{8}-h_{7}-2h_{6}-2h_{5}-2h_{4}-2h_{3}-h_{2}-h_{1}-g_{2}
(0, -1, -2, -2, -2, -2, -1, -1)-e_{2}-e_{3}g_{-54}00000000000000000000000000000000000000000000000-g_{-56}0000000g_{-55}-g_{-54}0g_{-54}0000000g_{-52}000000-g_{-50}g_{-49}00000-g_{-47}g_{-46}00000-g_{-43}g_{-42}0000-g_{-38}g_{-37}g_{-36}000-g_{-32}-g_{-31}g_{-30}000-g_{-25}g_{-24}00-g_{-18}g_{-17}0-g_{-11}g_{-10}0-g_{-3}0-h_{8}-h_{7}-2h_{6}-2h_{5}-2h_{4}-2h_{3}-h_{2}-g_{1}g_{9}
(-1, -1, -1, -2, -2, -2, -1, -1)-e_{1}-e_{4}g_{-53}0000000000000000000000000000000000000000000000g_{-56}000000g_{-55}00g_{-53}0-g_{-53}g_{-53}0000g_{-52}00g_{-51}0000g_{-50}00g_{-48}000000g_{-45}000-g_{-44}00g_{-41}g_{-40}0-g_{-39}000g_{-35}0-g_{-33}000g_{-29}-g_{-27}-g_{-26}00g_{-23}-g_{-19}00-g_{-12}0-g_{-9}-g_{-4}-g_{-1}-h_{8}-h_{7}-2h_{6}-2h_{5}-2h_{4}-h_{3}-h_{2}-h_{1}0-g_{3}-g_{10}
(0, -1, -1, -2, -2, -2, -1, -1)-e_{2}-e_{4}g_{-52}0000000000000000000000000000000000000000-g_{-56}000000000000g_{-54}0g_{-53}-g_{-52}g_{-52}-g_{-52}g_{-52}00000g_{-50}0g_{-49}0000000g_{-46}0000-g_{-44}0g_{-42}0000-g_{-39}0g_{-37}g_{-36}00-g_{-33}00g_{-30}00-g_{-27}-g_{-26}0g_{-24}00-g_{-19}0g_{-17}0-g_{-12}00-g_{-4}-g_{-2}0-h_{8}-h_{7}-2h_{6}-2h_{5}-2h_{4}-h_{3}-h_{2}-g_{1}-g_{3}0g_{16}
(-1, -1, -1, -1, -2, -2, -1, -1)-e_{1}-e_{5}g_{-51}000000000000000000000000000000000000000g_{-56}00000g_{-55}000000g_{-53}000g_{-51}00-g_{-51}g_{-51}000g_{-49}000g_{-48}000g_{-47}000g_{-45}00g_{-44}000g_{-41}g_{-40}000000g_{-35}-g_{-34}0000g_{-29}-g_{-28}0000-g_{-21}-g_{-20}00-g_{-16}-g_{-13}0-g_{-9}-g_{-5}-g_{-1}0-h_{8}-h_{7}-2h_{6}-2h_{5}-h_{4}-h_{3}-h_{2}-h_{1}0-g_{4}0-g_{11}-g_{17}
(0, 0, -1, -2, -2, -2, -1, -1)-e_{3}-e_{4}g_{-50}0000000000000000000000000000000000000000-g_{-55}00000-g_{-54}g_{-53}000000g_{-52}00-g_{-50}0g_{-50}0000000g_{-47}000000-g_{-44}g_{-43}00000-g_{-39}g_{-38}00000-g_{-33}g_{-32}g_{-31}000-g_{-27}-g_{-26}g_{-25}0000-g_{-19}g_{-18}000-g_{-12}g_{-11}00-g_{-4}00-h_{8}-h_{7}-2h_{6}-2h_{5}-2h_{4}-h_{3}0-g_{2}-g_{9}g_{10}g_{16}0
(0, -1, -1, -1, -2, -2, -1, -1)-e_{2}-e_{5}g_{-49}000000000000000000000000000000000-g_{-56}00000000000g_{-54}000000g_{-52}00g_{-51}-g_{-49}g_{-49}0-g_{-49}g_{-49}0000g_{-47}00g_{-46}0000g_{-44}00g_{-42}000000g_{-37}g_{-36}00-g_{-34}000g_{-30}0-g_{-28}000g_{-24}0-g_{-21}-g_{-20}0000-g_{-13}0-g_{-10}0-g_{-5}-g_{-2}0-h_{8}-h_{7}-2h_{6}-2h_{5}-h_{4}-h_{3}-h_{2}0-g_{1}-g_{4}0-g_{11}0g_{23}
(-1, -1, -1, -1, -1, -2, -1, -1)-e_{1}-e_{6}g_{-48}00000000000000000000000000000000g_{-56}00000g_{-55}00000g_{-53}000000g_{-51}0000g_{-48}000-g_{-48}g_{-48}00g_{-46}0000g_{-45}00g_{-43}0000g_{-41}g_{-40}g_{-39}00000g_{-35}g_{-34}0000000000-g_{-23}-g_{-22}000-g_{-16}-g_{-15}-g_{-14}0-g_{-9}0-g_{-6}-g_{-1}0-h_{8}-h_{7}-2h_{6}-h_{5}-h_{4}-h_{3}-h_{2}-h_{1}00-g_{5}0-g_{12}0-g_{18}-g_{24}
(0, 0, -1, -1, -2, -2, -1, -1)-e_{3}-e_{5}g_{-47}000000000000000000000000000000000-g_{-55}00000-g_{-54}0000000g_{-51}0000g_{-50}0g_{-49}00-g_{-47}g_{-47}-g_{-47}g_{-47}00000g_{-44}0g_{-43}0000000g_{-38}0000-g_{-34}0g_{-32}g_{-31}000-g_{-28}00g_{-25}00-g_{-21}-g_{-20}0g_{-18}000-g_{-13}0000-g_{-5}-g_{-3}00-h_{8}-h_{7}-2h_{6}-2h_{5}-h_{4}-h_{3}0-g_{2}-g_{4}-g_{9}00g_{17}g_{23}0
(0, -1, -1, -1, -1, -2, -1, -1)-e_{2}-e_{6}g_{-46}000000000000000000000000000-g_{-56}0000000000g_{-54}00000g_{-52}000000g_{-49}000g_{-48}-g_{-46}g_{-46}00-g_{-46}g_{-46}000g_{-43}000g_{-42}000g_{-39}000g_{-37}g_{-36}0g_{-34}0000g_{-30}0000000-g_{-22}000-g_{-17}0-g_{-15}-g_{-14}0-g_{-10}00-g_{-6}-g_{-2}00-h_{8}-h_{7}-2h_{6}-h_{5}-h_{4}-h_{3}-h_{2}0-g_{1}-g_{5}00-g_{12}0-g_{18}0g_{29}
(-1, -1, -1, -1, -1, -1, -1, -1)-e_{1}-e_{7}g_{-45}00000000000000000000000000g_{-56}0000g_{-55}00000g_{-53}00000g_{-51}000000g_{-48}00000g_{-45}0000-g_{-45}g_{-45}g_{-45}g_{-42}00000g_{-41}g_{-40}g_{-38}000000g_{-33}00000-g_{-29}g_{-28}0000-g_{-23}g_{-22}000-g_{-16}0000-g_{-9}0-g_{-8}-g_{-7}-g_{-1}00-h_{8}-h_{7}-h_{6}-h_{5}-h_{4}-h_{3}-h_{2}-h_{1}00-g_{6}00-g_{13}0-g_{19}0-g_{25}-g_{30}
(0, 0, 0, -1, -2, -2, -1, -1)-e_{4}-e_{5}g_{-44}000000000000000000000000000000000-g_{-53}00000-g_{-52}g_{-51}0000-g_{-50}g_{-49}000000g_{-47}0000-g_{-44}0g_{-44}0000000g_{-39}000000-g_{-34}g_{-33}00000-g_{-28}g_{-27}g_{-26}0000-g_{-21}-g_{-20}g_{-19}0000-g_{-13}g_{-12}0000-g_{-5}0000-h_{8}-h_{7}-2h_{6}-2h_{5}-h_{4}00-g_{3}0-g_{10}g_{11}-g_{16}g_{17}g_{23}000
(0, 0, -1, -1, -1, -2, -1, -1)-e_{3}-e_{6}g_{-43}000000000000000000000000000-g_{-55}0000-g_{-54}00000000000g_{-50}00g_{-48}000g_{-47}00g_{-46}00-g_{-43}g_{-43}0-g_{-43}g_{-43}0000g_{-39}00g_{-38}0000g_{-34}00g_{-32}g_{-31}000000g_{-25}00-g_{-22}00000-g_{-15}-g_{-14}0-g_{-11}000-g_{-6}-g_{-3}000-h_{8}-h_{7}-2h_{6}-h_{5}-h_{4}-h_{3}00-g_{2}-g_{5}-g_{9}0-g_{12}000g_{24}g_{29}0
(0, -1, -1, -1, -1, -1, -1, -1)-e_{2}-e_{7}g_{-42}000000000000000000000-g_{-56}000000000g_{-54}00000g_{-52}00000g_{-49}000000g_{-46}0000g_{-45}-g_{-42}g_{-42}000-g_{-42}g_{-42}g_{-42}0g_{-38}0000g_{-37}g_{-36}0g_{-33}000000g_{-28}0000-g_{-24}0g_{-22}000-g_{-17}0000-g_{-10}00-g_{-8}-g_{-7}-g_{-2}000-h_{8}-h_{7}-h_{6}-h_{5}-h_{4}-h_{3}-h_{2}00-g_{1}-g_{6}00-g_{13}00-g_{19}0-g_{25}0g_{35}
(-1, -1, -1, -1, -1, -1, 0, -1)-e_{1}-e_{8}g_{-41}00000000000000000000g_{-56}0000g_{-55}0000g_{-53}00000g_{-51}00000g_{-48}000000g_{-45}000000g_{-41}00000-g_{-41}g_{-41}g_{-37}000000-g_{-35}g_{-32}00000-g_{-29}g_{-27}0000-g_{-23}0g_{-21}000-g_{-16}0g_{-15}00-g_{-9}00g_{-8}0-g_{-1}000-h_{8}-h_{6}-h_{5}-h_{4}-h_{3}-h_{2}-h_{1}000-g_{7}00-g_{14}00-g_{20}0-g_{26}0-g_{31}-g_{36}
(-1, -1, -1, -1, -1, -1, -1, 0)-e_{1}+e_{8}g_{-40}0000000000000000000g_{-56}0000g_{-55}0000g_{-53}00000g_{-51}00000g_{-48}000000g_{-45}0000000g_{-40}00000g_{-40}-g_{-40}g_{-36}00000-g_{-35}0g_{-31}0000-g_{-29}0g_{-26}000-g_{-23}00g_{-20}00-g_{-16}00g_{-14}0-g_{-9}000g_{-7}-g_{-1}000-h_{7}-h_{6}-h_{5}-h_{4}-h_{3}-h_{2}-h_{1}0000-g_{8}00-g_{15}00-g_{21}0-g_{27}0-g_{32}-g_{37}
(0, 0, 0, -1, -1, -2, -1, -1)-e_{4}-e_{6}g_{-39}000000000000000000000000000-g_{-53}0000-g_{-52}00000-g_{-50}0g_{-48}00000g_{-46}0000g_{-44}0g_{-43}0000-g_{-39}g_{-39}-g_{-39}g_{-39}00000g_{-34}0g_{-33}0000000g_{-27}g_{-26}000-g_{-22}00g_{-19}000-g_{-15}-g_{-14}00000-g_{-6}-g_{-4}0000-h_{8}-h_{7}-2h_{6}-h_{5}-h_{4}000-g_{3}-g_{5}0-g_{10}0-g_{16}0g_{18}0g_{24}g_{29}000
(0, 0, -1, -1, -1, -1, -1, -1)-e_{3}-e_{7}g_{-38}000000000000000000000-g_{-55}0000-g_{-54}0000000000g_{-50}00000g_{-47}000g_{-45}00g_{-43}000g_{-42}00-g_{-38}g_{-38}00-g_{-38}g_{-38}g_{-38}00g_{-33}000g_{-32}g_{-31}00g_{-28}000000g_{-22}000-g_{-18}00000-g_{-11}00-g_{-8}-g_{-7}-g_{-3}0000-h_{8}-h_{7}-h_{6}-h_{5}-h_{4}-h_{3}000-g_{2}-g_{6}0-g_{9}0-g_{13}00-g_{19}000g_{30}g_{35}0
(0, -1, -1, -1, -1, -1, 0, -1)-e_{2}-e_{8}g_{-37}0000000000000000-g_{-56}00000000g_{-54}0000g_{-52}00000g_{-49}00000g_{-46}000000g_{-42}00000g_{-41}-g_{-37}g_{-37}0000-g_{-37}g_{-37}0g_{-32}00000-g_{-30}0g_{-27}0000-g_{-24}0g_{-21}000-g_{-17}00g_{-15}00-g_{-10}00g_{-8}0-g_{-2}0000-h_{8}-h_{6}-h_{5}-h_{4}-h_{3}-h_{2}000-g_{1}-g_{7}000-g_{14}00-g_{20}00-g_{26}0-g_{31}0g_{40}
(0, -1, -1, -1, -1, -1, -1, 0)-e_{2}+e_{8}g_{-36}000000000000000-g_{-56}00000000g_{-54}0000g_{-52}00000g_{-49}00000g_{-46}000000g_{-42}000000g_{-40}-g_{-36}g_{-36}0000g_{-36}-g_{-36}0g_{-31}0000-g_{-30}00g_{-26}000-g_{-24}00g_{-20}00-g_{-17}000g_{-14}0-g_{-10}000g_{-7}-g_{-2}0000-h_{7}-h_{6}-h_{5}-h_{4}-h_{3}-h_{2}000-g_{1}0-g_{8}000-g_{15}00-g_{21}00-g_{27}0-g_{32}0g_{41}
(-1, -1, -1, -1, -1, -1, 0, 0)-e_{1}+e_{7}g_{-35}00000000000000g_{-56}000g_{-55}0000g_{-53}0000g_{-51}00000g_{-48}0000000000000-g_{-41}-g_{-40}000000g_{-35}0000g_{-35}-g_{-35}-g_{-35}g_{-30}0000-g_{-29}00g_{-25}000-g_{-23}00g_{-19}00-g_{-16}000g_{-13}0-g_{-9}000g_{-6}-g_{-1}0000-h_{6}-h_{5}-h_{4}-h_{3}-h_{2}-h_{1}0000g_{7}g_{8}000000-g_{22}00-g_{28}0-g_{33}0-g_{38}-g_{42}
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(0, 0, 1, 1, 1, 1, 1, 1)e_{3}+e_{7}g_{38}0-g_{-35}-g_{-30}000g_{-19}00g_{-13}0g_{-9}0g_{-6}g_{-2}000h_{8}+h_{7}+h_{6}+h_{5}+h_{4}+h_{3}0000g_{3}g_{7}g_{8}00g_{11}00000g_{18}000-g_{22}000000-g_{28}00-g_{31}-g_{32}000-g_{33}000g_{38}-g_{38}00g_{38}-g_{38}-g_{38}0-g_{42}000-g_{43}00-g_{45}000-g_{47}00000-g_{50}0000000000g_{54}0000g_{55}000000000000000000000
(0, 0, 0, 1, 1, 2, 1, 1)e_{4}+e_{6}g_{39}000-g_{-29}-g_{-24}0-g_{-18}0g_{-16}0g_{-10}0g_{-5}g_{-3}000h_{8}+h_{7}+2h_{6}+h_{5}+h_{4}0000g_{4}g_{6}00000g_{14}g_{15}000-g_{19}00g_{22}000-g_{26}-g_{27}0000000-g_{33}0-g_{34}00000g_{39}-g_{39}g_{39}-g_{39}0000-g_{43}0-g_{44}0000-g_{46}00000-g_{48}0g_{50}00000g_{52}0000g_{53}000000000000000000000000000
(1, 1, 1, 1, 1, 1, 1, 0)e_{1}-e_{8}g_{40}g_{-37}g_{-32}0g_{-27}0g_{-21}00g_{-15}00g_{-8}0000h_{7}+h_{6}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}000g_{1}-g_{7}000g_{9}0-g_{14}00g_{16}00-g_{20}00g_{23}000-g_{26}0g_{29}0000-g_{31}0g_{35}00000-g_{36}-g_{40}00000-g_{40}g_{40}0000000-g_{45}000000-g_{48}00000-g_{51}00000-g_{53}0000-g_{55}0000-g_{56}0000000000000000000
(1, 1, 1, 1, 1, 1, 0, 1)e_{1}+e_{8}g_{41}g_{-36}g_{-31}0g_{-26}0g_{-20}00g_{-14}00g_{-7}000h_{8}+h_{6}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}000g_{1}0-g_{8}00g_{9}00-g_{15}0g_{16}000-g_{21}0g_{23}0000-g_{27}g_{29}00000-g_{32}g_{35}000000-g_{37}-g_{41}00000g_{41}-g_{41}000000-g_{45}000000-g_{48}00000-g_{51}00000-g_{53}0000-g_{55}0000-g_{56}00000000000000000000
(0, 1, 1, 1, 1, 1, 1, 1)e_{2}+e_{7}g_{42}-g_{-35}0g_{-25}0g_{-19}00g_{-13}00g_{-6}g_{-1}00h_{8}+h_{7}+h_{6}+h_{5}+h_{4}+h_{3}+h_{2}000g_{2}g_{7}g_{8}00g_{10}0000g_{17}000-g_{22}0g_{24}0000-g_{28}000000-g_{33}0-g_{36}-g_{37}0000-g_{38}0g_{42}-g_{42}000g_{42}-g_{42}-g_{42}-g_{45}0000-g_{46}000000-g_{49}00000-g_{52}00000-g_{54}000000000g_{56}000000000000000000000
(0, 0, 1, 1, 1, 2, 1, 1)e_{3}+e_{6}g_{43}0-g_{-29}-g_{-24}000g_{-12}0g_{-9}g_{-5}g_{-2}00h_{8}+h_{7}+2h_{6}+h_{5}+h_{4}+h_{3}000g_{3}g_{6}000g_{11}0g_{14}g_{15}00000g_{22}00-g_{25}000000-g_{31}-g_{32}00-g_{34}0000-g_{38}00-g_{39}000g_{43}-g_{43}0g_{43}-g_{43}000-g_{46}00-g_{47}000-g_{48}00-g_{50}00000000000g_{54}0000g_{55}000000000000000000000000000
(0, 0, 0, 1, 2, 2, 1, 1)e_{4}+e_{5}g_{44}000-g_{-23}-g_{-17}g_{-16}-g_{-11}g_{-10}0g_{-3}00h_{8}+h_{7}+2h_{6}+2h_{5}+h_{4}0000g_{5}0000-g_{12}g_{13}0000-g_{19}g_{20}g_{21}0000-g_{26}-g_{27}g_{28}00000-g_{33}g_{34}000000-g_{39}000000g_{44}0-g_{44}00000-g_{47}000000-g_{49}g_{50}0000-g_{51}g_{52}00000g_{53}000000000000000000000000000000000
(1, 1, 1, 1, 1, 1, 1, 1)e_{1}+e_{7}g_{45}g_{-30}g_{-25}0g_{-19}0g_{-13}00g_{-6}00h_{8}+h_{7}+h_{6}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}00g_{1}g_{7}g_{8}0g_{9}0000g_{16}000-g_{22}g_{23}0000-g_{28}g_{29}00000-g_{33}000000-g_{38}-g_{40}-g_{41}00000-g_{42}-g_{45}0000g_{45}-g_{45}-g_{45}00000-g_{48}000000-g_{51}00000-g_{53}00000-g_{55}0000-g_{56}00000000000000000000000000
(0, 1, 1, 1, 1, 2, 1, 1)e_{2}+e_{6}g_{46}-g_{-29}0g_{-18}0g_{-12}00g_{-5}g_{-1}0h_{8}+h_{7}+2h_{6}+h_{5}+h_{4}+h_{3}+h_{2}00g_{2}g_{6}00g_{10}0g_{14}g_{15}0g_{17}000g_{22}0000000-g_{30}0000-g_{34}0-g_{36}-g_{37}000-g_{39}000-g_{42}000-g_{43}0g_{46}-g_{46}00g_{46}-g_{46}00-g_{48}000-g_{49}000000-g_{52}00000-g_{54}0000000000g_{56}000000000000000000000000000
(0, 0, 1, 1, 2, 2, 1, 1)e_{3}+e_{5}g_{47}0-g_{-23}-g_{-17}00g_{-9}g_{-4}g_{-2}0h_{8}+h_{7}+2h_{6}+2h_{5}+h_{4}+h_{3}00g_{3}g_{5}0000g_{13}000-g_{18}0g_{20}g_{21}00-g_{25}00g_{28}000-g_{31}-g_{32}0g_{34}0000-g_{38}0000000-g_{43}0-g_{44}000g_{47}-g_{47}g_{47}-g_{47}0000-g_{49}0-g_{50}0000-g_{51}0000000g_{54}00000g_{55}000000000000000000000000000000000
(1, 1, 1, 1, 1, 2, 1, 1)e_{1}+e_{6}g_{48}g_{-24}g_{-18}0g_{-12}0g_{-5}00h_{8}+h_{7}+2h_{6}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}0g_{1}g_{6}0g_{9}0g_{14}g_{15}g_{16}000g_{22}g_{23}0000000000-g_{34}-g_{35}00000-g_{39}-g_{40}-g_{41}0000-g_{43}00-g_{45}0000-g_{46}-g_{48}000g_{48}-g_{48}000000-g_{51}000000-g_{53}00000-g_{55}00000-g_{56}00000000000000000000000000000000
(0, 1, 1, 1, 2, 2, 1, 1)e_{2}+e_{5}g_{49}-g_{-23}0g_{-11}0g_{-4}g_{-1}0h_{8}+h_{7}+2h_{6}+2h_{5}+h_{4}+h_{3}+h_{2}0g_{2}g_{5}0g_{10}0g_{13}0000g_{20}g_{21}0-g_{24}000g_{28}0-g_{30}000g_{34}00-g_{36}-g_{37}000000-g_{42}00-g_{44}0000-g_{46}00-g_{47}0g_{49}-g_{49}0g_{49}-g_{49}000-g_{51}00-g_{52}000000-g_{54}00000000000g_{56}000000000000000000000000000000000
(0, 0, 1, 2, 2, 2, 1, 1)e_{3}+e_{4}g_{50}0-g_{-16}-g_{-10}g_{-9}g_{-2}0h_{8}+h_{7}+2h_{6}+2h_{5}+2h_{4}+h_{3}00g_{4}00-g_{11}g_{12}000-g_{18}g_{19}0000-g_{25}g_{26}g_{27}000-g_{31}-g_{32}g_{33}00000-g_{38}g_{39}00000-g_{43}g_{44}000000-g_{47}0000g_{50}0-g_{50}00000-g_{52}000000-g_{53}g_{54}00000g_{55}0000000000000000000000000000000000000000
(1, 1, 1, 1, 2, 2, 1, 1)e_{1}+e_{5}g_{51}g_{-17}g_{-11}0g_{-4}0h_{8}+h_{7}+2h_{6}+2h_{5}+h_{4}+h_{3}+h_{2}+h_{1}0g_{1}g_{5}g_{9}0g_{13}g_{16}00g_{20}g_{21}0000g_{28}-g_{29}0000g_{34}-g_{35}000000-g_{40}-g_{41}000-g_{44}00-g_{45}000-g_{47}000-g_{48}000-g_{49}-g_{51}00g_{51}-g_{51}000000-g_{53}000000-g_{55}00000-g_{56}000000000000000000000000000000000000000
(0, 1, 1, 2, 2, 2, 1, 1)e_{2}+e_{4}g_{52}-g_{-16}0g_{-3}g_{-1}h_{8}+h_{7}+2h_{6}+2h_{5}+2h_{4}+h_{3}+h_{2}0g_{2}g_{4}00g_{12}0-g_{17}0g_{19}00-g_{24}0g_{26}g_{27}00-g_{30}00g_{33}00-g_{36}-g_{37}0g_{39}0000-g_{42}0g_{44}0000-g_{46}0000000-g_{49}0-g_{50}0g_{52}-g_{52}g_{52}-g_{52}0000-g_{53}0-g_{54}000000000000g_{56}0000000000000000000000000000000000000000
(1, 1, 1, 2, 2, 2, 1, 1)e_{1}+e_{4}g_{53}g_{-10}g_{-3}0h_{8}+h_{7}+2h_{6}+2h_{5}+2h_{4}+h_{3}+h_{2}+h_{1}g_{1}g_{4}g_{9}0g_{12}00g_{19}-g_{23}00g_{26}g_{27}-g_{29}000g_{33}0-g_{35}000g_{39}0-g_{40}-g_{41}00g_{44}000-g_{45}000000-g_{48}00-g_{50}0000-g_{51}00-g_{52}-g_{53}0g_{53}-g_{53}000000-g_{55}000000-g_{56}0000000000000000000000000000000000000000000000
(0, 1, 2, 2, 2, 2, 1, 1)e_{2}+e_{3}g_{54}-g_{-9}g_{-1}h_{8}+h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+h_{2}0g_{3}0-g_{10}g_{11}0-g_{17}g_{18}00-g_{24}g_{25}000-g_{30}g_{31}g_{32}000-g_{36}-g_{37}g_{38}0000-g_{42}g_{43}00000-g_{46}g_{47}00000-g_{49}g_{50}000000-g_{52}00g_{54}0-g_{54}00000-g_{55}0000000g_{56}00000000000000000000000000000000000000000000000
(1, 1, 2, 2, 2, 2, 1, 1)e_{1}+e_{3}g_{55}g_{-2}h_{8}+h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+h_{2}+h_{1}g_{1}g_{3}0g_{11}-g_{16}0g_{18}-g_{23}0g_{25}0-g_{29}0g_{31}g_{32}0-g_{35}00g_{38}00-g_{40}-g_{41}0g_{43}000-g_{45}0g_{47}0000-g_{48}0g_{50}0000-g_{51}0000000-g_{53}0-g_{54}-g_{55}g_{55}-g_{55}000000-g_{56}000000000000000000000000000000000000000000000000000000
(1, 2, 2, 2, 2, 2, 1, 1)e_{1}+e_{2}g_{56}h_{8}+h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}g_{2}-g_{9}g_{10}-g_{16}g_{17}0-g_{23}g_{24}0-g_{29}g_{30}00-g_{35}g_{36}g_{37}00-g_{40}-g_{41}g_{42}0000-g_{45}g_{46}0000-g_{48}g_{49}00000-g_{51}g_{52}00000-g_{53}g_{54}000000-g_{55}00-g_{56}00000000000000000000000000000000000000000000000000000000000000
We define the symmetric Cartan matrix
by requesting that the entry in the i-th row and j-th column
be the scalar product of the i^th and j^th roots. The symmetric Cartan matrix is:
\(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & -1\\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 2\\ \end{pmatrix}\)
Let the (i, j)^{th} entry of the symmetric Cartan matrix be a_{ij}.
Then we define the co-symmetric Cartan matrix as the matrix whose (i, j)^{th} entry equals 4*a_{ij}/(a_{ii}*a_{jj}). In other words, the co-symmetric Cartan matrix is the symmetric Cartan matrix of the dual root system. The co-symmetric Cartan matrix equals:
\(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & -1\\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 2\\ \end{pmatrix}\)
The determinant of the symmetric Cartan matrix is: 4
Half sum of positive roots: (7, 13, 18, 22, 25, 27, 14, 14)= \(\displaystyle 7\varepsilon_{1}+6\varepsilon_{2}+5\varepsilon_{3}+4\varepsilon_{4}+3\varepsilon_{5}+2\varepsilon_{6}+\varepsilon_{7}\)
The fundamental weights (the j^th fundamental weight has scalar product 1
with the j^th simple root times 2 divided by the root length squared,
and 0 with the remaining simple roots):
(1, 1, 1, 1, 1, 1, 1/2, 1/2) = \(\displaystyle \varepsilon_{1}\)
(1, 2, 2, 2, 2, 2, 1, 1) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}\)
(1, 2, 3, 3, 3, 3, 3/2, 3/2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}\)
(1, 2, 3, 4, 4, 4, 2, 2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}\)
(1, 2, 3, 4, 5, 5, 5/2, 5/2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}\)
(1, 2, 3, 4, 5, 6, 3, 3) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}+\varepsilon_{6}\)
(1/2, 1, 3/2, 2, 5/2, 3, 2, 3/2) = \(\displaystyle 1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}\)
(1/2, 1, 3/2, 2, 5/2, 3, 3/2, 2) = \(\displaystyle 1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}+1/2\varepsilon_{8}\)
Below is the simple basis realized in epsilon coordinates. Please note that the epsilon coordinate realizations do not have long roots of length of 2 in types G and C. This means that gramm matrix (w.r.t. the standard scalar product) of the epsilon coordinate realizations in types G and C does not equal the corresponding symmetric Cartan matrix.
(1, 0, 0, 0, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{1}-\varepsilon_{2}\)
(0, 1, 0, 0, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{2}-\varepsilon_{3}\)
(0, 0, 1, 0, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{3}-\varepsilon_{4}\)
(0, 0, 0, 1, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{4}-\varepsilon_{5}\)
(0, 0, 0, 0, 1, 0, 0, 0) = \(\displaystyle \varepsilon_{5}-\varepsilon_{6}\)
(0, 0, 0, 0, 0, 1, 0, 0) = \(\displaystyle \varepsilon_{6}-\varepsilon_{7}\)
(0, 0, 0, 0, 0, 0, 1, 0) = \(\displaystyle \varepsilon_{7}-\varepsilon_{8}\)
(0, 0, 0, 0, 0, 0, 0, 1) = \(\displaystyle \varepsilon_{7}+\varepsilon_{8}\)