\(E^{1}_6\)
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Semisimple subalgebras.
Page generated by the calculator project.
Lie algebra type: E^{1}_6.
Weyl group size: 51840.
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 72 elements.
Simple basis coordinatesEpsilon coordinatesReflection w.r.t. root
(-1, -2, -2, -3, -2, -1)1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{2}s_{4}s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}\)
(-1, -1, -2, -3, -2, -1)-1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{4}s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}\)
(-1, -1, -2, -2, -2, -1)-1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}\)
(-1, -1, -1, -2, -2, -1)1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{5}\)
(-1, -1, -2, -2, -1, -1)-1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{3}s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}\)
(0, -1, -1, -2, -2, -1)e_{4}+e_{5}\(s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}\)
(-1, -1, -1, -2, -1, -1)1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}\)
(-1, -1, -2, -2, -1, 0)-1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{3}s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}\)
(0, -1, -1, -2, -1, -1)e_{3}+e_{5}\(s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}\)
(-1, -1, -1, -1, -1, -1)1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}\)
(-1, -1, -1, -2, -1, 0)1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}\)
(0, -1, -1, -1, -1, -1)e_{2}+e_{5}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}\)
(-1, 0, -1, -1, -1, -1)-1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{1}\)
(0, -1, -1, -2, -1, 0)e_{3}+e_{4}\(s_{4}s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}s_{4}\)
(-1, -1, -1, -1, -1, 0)1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}s_{1}\)
(0, 0, -1, -1, -1, -1)-e_{1}+e_{5}\(s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, 0, -1, -1, -1)e_{1}+e_{5}\(s_{2}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}\)
(0, -1, -1, -1, -1, 0)e_{2}+e_{4}\(s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}\)
(-1, 0, -1, -1, -1, 0)-1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{3}s_{4}s_{5}s_{4}s_{3}s_{1}\)
(-1, -1, -1, -1, 0, 0)1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{2}s_{3}s_{1}\)
(0, 0, 0, -1, -1, -1)-e_{2}+e_{5}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(0, 0, -1, -1, -1, 0)-e_{1}+e_{4}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, -1, 0, -1, -1, 0)e_{1}+e_{4}\(s_{2}s_{4}s_{5}s_{4}s_{2}\)
(0, -1, -1, -1, 0, 0)e_{2}+e_{3}\(s_{2}s_{3}s_{4}s_{2}s_{3}\)
(-1, 0, -1, -1, 0, 0)-1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{3}s_{4}s_{3}s_{1}\)
(0, 0, 0, 0, -1, -1)-e_{3}+e_{5}\(s_{5}s_{6}s_{5}\)
(0, 0, 0, -1, -1, 0)-e_{2}+e_{4}\(s_{4}s_{5}s_{4}\)
(0, 0, -1, -1, 0, 0)-e_{1}+e_{3}\(s_{3}s_{4}s_{3}\)
(0, -1, 0, -1, 0, 0)e_{1}+e_{3}\(s_{2}s_{4}s_{2}\)
(-1, 0, -1, 0, 0, 0)-1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}s_{3}s_{1}\)
(0, 0, 0, 0, 0, -1)-e_{4}+e_{5}\(s_{6}\)
(0, 0, 0, 0, -1, 0)-e_{3}+e_{4}\(s_{5}\)
(0, 0, 0, -1, 0, 0)-e_{2}+e_{3}\(s_{4}\)
(0, 0, -1, 0, 0, 0)-e_{1}+e_{2}\(s_{3}\)
(0, -1, 0, 0, 0, 0)e_{1}+e_{2}\(s_{2}\)
(-1, 0, 0, 0, 0, 0)1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}\(s_{1}\)
(1, 0, 0, 0, 0, 0)-1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}\)
(0, 1, 0, 0, 0, 0)-e_{1}-e_{2}\(s_{2}\)
(0, 0, 1, 0, 0, 0)e_{1}-e_{2}\(s_{3}\)
(0, 0, 0, 1, 0, 0)e_{2}-e_{3}\(s_{4}\)
(0, 0, 0, 0, 1, 0)e_{3}-e_{4}\(s_{5}\)
(0, 0, 0, 0, 0, 1)e_{4}-e_{5}\(s_{6}\)
(1, 0, 1, 0, 0, 0)1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{3}s_{1}\)
(0, 1, 0, 1, 0, 0)-e_{1}-e_{3}\(s_{2}s_{4}s_{2}\)
(0, 0, 1, 1, 0, 0)e_{1}-e_{3}\(s_{3}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 0)e_{2}-e_{4}\(s_{4}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1)e_{3}-e_{5}\(s_{5}s_{6}s_{5}\)
(1, 0, 1, 1, 0, 0)1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{3}s_{4}s_{3}s_{1}\)
(0, 1, 1, 1, 0, 0)-e_{2}-e_{3}\(s_{2}s_{3}s_{4}s_{2}s_{3}\)
(0, 1, 0, 1, 1, 0)-e_{1}-e_{4}\(s_{2}s_{4}s_{5}s_{4}s_{2}\)
(0, 0, 1, 1, 1, 0)e_{1}-e_{4}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1)e_{2}-e_{5}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(1, 1, 1, 1, 0, 0)-1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{2}s_{3}s_{1}\)
(1, 0, 1, 1, 1, 0)1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{3}s_{4}s_{5}s_{4}s_{3}s_{1}\)
(0, 1, 1, 1, 1, 0)-e_{2}-e_{4}\(s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}\)
(0, 1, 0, 1, 1, 1)-e_{1}-e_{5}\(s_{2}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}\)
(0, 0, 1, 1, 1, 1)e_{1}-e_{5}\(s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}\)
(1, 1, 1, 1, 1, 0)-1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}s_{1}\)
(0, 1, 1, 2, 1, 0)-e_{3}-e_{4}\(s_{4}s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}s_{4}\)
(1, 0, 1, 1, 1, 1)1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{1}\)
(0, 1, 1, 1, 1, 1)-e_{2}-e_{5}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}\)
(1, 1, 1, 2, 1, 0)-1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}\)
(1, 1, 1, 1, 1, 1)-1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}\)
(0, 1, 1, 2, 1, 1)-e_{3}-e_{5}\(s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}\)
(1, 1, 2, 2, 1, 0)1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{3}s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}\)
(1, 1, 1, 2, 1, 1)-1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}\)
(0, 1, 1, 2, 2, 1)-e_{4}-e_{5}\(s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}\)
(1, 1, 2, 2, 1, 1)1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{3}s_{1}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}\)
(1, 1, 1, 2, 2, 1)-1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{5}\)
(1, 1, 2, 2, 2, 1)1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}\)
(1, 1, 2, 3, 2, 1)1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{4}s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}\)
(1, 2, 2, 3, 2, 1)-1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}\(s_{2}s_{4}s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}\)
Comma delimited list of roots: (-1, -2, -2, -3, -2, -1), (-1, -1, -2, -3, -2, -1), (-1, -1, -2, -2, -2, -1), (-1, -1, -1, -2, -2, -1), (-1, -1, -2, -2, -1, -1), (0, -1, -1, -2, -2, -1), (-1, -1, -1, -2, -1, -1), (-1, -1, -2, -2, -1, 0), (0, -1, -1, -2, -1, -1), (-1, -1, -1, -1, -1, -1), (-1, -1, -1, -2, -1, 0), (0, -1, -1, -1, -1, -1), (-1, 0, -1, -1, -1, -1), (0, -1, -1, -2, -1, 0), (-1, -1, -1, -1, -1, 0), (0, 0, -1, -1, -1, -1), (0, -1, 0, -1, -1, -1), (0, -1, -1, -1, -1, 0), (-1, 0, -1, -1, -1, 0), (-1, -1, -1, -1, 0, 0), (0, 0, 0, -1, -1, -1), (0, 0, -1, -1, -1, 0), (0, -1, 0, -1, -1, 0), (0, -1, -1, -1, 0, 0), (-1, 0, -1, -1, 0, 0), (0, 0, 0, 0, -1, -1), (0, 0, 0, -1, -1, 0), (0, 0, -1, -1, 0, 0), (0, -1, 0, -1, 0, 0), (-1, 0, -1, 0, 0, 0), (0, 0, 0, 0, 0, -1), (0, 0, 0, 0, -1, 0), (0, 0, 0, -1, 0, 0), (0, 0, -1, 0, 0, 0), (0, -1, 0, 0, 0, 0), (-1, 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), (1, 0, 1, 0, 0, 0), (0, 1, 0, 1, 0, 0), (0, 0, 1, 1, 0, 0), (0, 0, 0, 1, 1, 0), (0, 0, 0, 0, 1, 1), (1, 0, 1, 1, 0, 0), (0, 1, 1, 1, 0, 0), (0, 1, 0, 1, 1, 0), (0, 0, 1, 1, 1, 0), (0, 0, 0, 1, 1, 1), (1, 1, 1, 1, 0, 0), (1, 0, 1, 1, 1, 0), (0, 1, 1, 1, 1, 0), (0, 1, 0, 1, 1, 1), (0, 0, 1, 1, 1, 1), (1, 1, 1, 1, 1, 0), (0, 1, 1, 2, 1, 0), (1, 0, 1, 1, 1, 1), (0, 1, 1, 1, 1, 1), (1, 1, 1, 2, 1, 0), (1, 1, 1, 1, 1, 1), (0, 1, 1, 2, 1, 1), (1, 1, 2, 2, 1, 0), (1, 1, 1, 2, 1, 1), (0, 1, 1, 2, 2, 1), (1, 1, 2, 2, 1, 1), (1, 1, 1, 2, 2, 1), (1, 1, 2, 2, 2, 1), (1, 1, 2, 3, 2, 1), (1, 2, 2, 3, 2, 1) The resulting Lie bracket pairing table follows.
Type E^{1}_6.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_{-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}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}
(-1, -2, -2, -3, -2, -1)1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}g_{-36}0000000000000000000000000000000000000g_{-36}00000g_{-35}00000g_{-34}0000-g_{-33}g_{-32}00g_{-31}0-g_{-30}-g_{-29}0g_{-28}g_{-27}0g_{-26}-g_{-25}-g_{-23}-g_{-22}g_{-20}g_{-19}-g_{-17}-g_{-14}g_{-13}-g_{-8}-g_{-2}-h_{6}-2h_{5}-3h_{4}-2h_{3}-2h_{2}-h_{1}
(-1, -1, -2, -3, -2, -1)-1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}g_{-35}0000000000000000000000000000000000g_{-36}00-g_{-35}0g_{-35}00000g_{-34}0000-g_{-33}g_{-32}0g_{-31}00-g_{-30}-g_{-29}0g_{-28}00g_{-26}0g_{-24}-g_{-23}0-g_{-21}0-g_{-18}g_{-16}g_{-15}-g_{-12}-g_{-10}g_{-9}-g_{-4}-h_{6}-2h_{5}-3h_{4}-2h_{3}-h_{2}-h_{1}-g_{2}
(-1, -1, -2, -2, -2, -1)-1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}-1/2e_{6}-1/2e_{7}+1/2e_{8}g_{-34}0000000000000000000000000000g_{-36}000g_{-35}00000g_{-34}-g_{-34}g_{-34}000g_{-33}0g_{-32}0-g_{-31}000-g_{-29}000-g_{-27}00g_{-25}g_{-24}0g_{-22}-g_{-21}0-g_{-19}-g_{-18}0g_{-15}0g_{-11}0g_{-7}-g_{-5}-g_{-3}-h_{6}-2h_{5}-2h_{4}-2h_{3}-h_{2}-h_{1}-g_{4}-g_{8}
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(0, 0, 0, 0, 0, 0)0h_{6}0000-g_{-32}0-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}0-g_{-16}g_{-15}g_{-14}00-g_{-11}g_{-10}000-2g_{-6}g_{-5}00000000000000-g_{5}2g_{6}000-g_{10}g_{11}00-g_{14}-g_{15}g_{16}0-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}0g_{32}0000
(1, 0, 0, 0, 0, 0)-1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}g_{1}000-g_{-31}00-g_{-28}00-g_{-25}-g_{-23}0-g_{-21}0-g_{-19}000-g_{-15}-g_{-13}0000-g_{-9}0000-g_{-3}00000h_{1}-2g_{1}0g_{1}00000g_{7}00000g_{12}000g_{17}0g_{18}000g_{22}0g_{24}0g_{26}0g_{27}00g_{30}00g_{33}00000
(0, 1, 0, 0, 0, 0)-e_{1}-e_{2}g_{2}-g_{-35}00000000-g_{-24}0-g_{-21}00-g_{-18}0-g_{-16}-g_{-15}0-g_{-12}00-g_{-10}-g_{-9}0000-g_{-4}00000h_{2}00-2g_{2}0g_{2}00000g_{8}0000g_{13}g_{14}0g_{17}00g_{19}g_{20}0g_{22}00g_{25}00g_{27}0000000000g_{36}0
(0, 0, 1, 0, 0, 0)e_{1}-e_{2}g_{3}00-g_{-33}0-g_{-30}00-g_{-26}000-g_{-20}000-g_{-16}0-g_{-14}000-g_{-10}0-g_{-8}000-g_{-4}0g_{-1}000h_{3}00g_{3}0-2g_{3}g_{3}00-g_{7}00g_{9}000g_{13}0g_{15}000g_{19}0g_{21}000g_{25}00000g_{29}000g_{32}00g_{34}000
(0, 0, 0, 1, 0, 0)e_{2}-e_{3}g_{4}0-g_{-34}0000-g_{-27}0-g_{-25}0-g_{-22}00-g_{-19}000000-g_{-11}000g_{-7}0-g_{-5}g_{-3}g_{-2}000h_{4}0000g_{4}g_{4}-2g_{4}g_{4}00-g_{8}-g_{9}0g_{10}0-g_{12}000g_{16}0000000g_{23}00g_{26}00g_{28}0g_{30}000000g_{35}00
(0, 0, 0, 0, 1, 0)e_{3}-e_{4}g_{5}00-g_{-32}-g_{-30}0-g_{-28}00000000g_{-17}00g_{-13}g_{-12}00g_{-9}g_{-8}00-g_{-6}g_{-4}0000h_{5}0000000g_{5}-2g_{5}g_{5}000-g_{10}0g_{11}0-g_{14}-g_{15}00-g_{18}-g_{19}000-g_{22}0000000000g_{31}0g_{33}0g_{34}0000
(0, 0, 0, 0, 0, 1)e_{4}-e_{5}g_{6}0000g_{-29}0g_{-26}0g_{-23}g_{-22}0g_{-19}g_{-18}00g_{-15}g_{-14}000g_{-10}0000g_{-5}0000h_{6}000000000g_{6}-2g_{6}0000-g_{11}0000-g_{16}000-g_{20}-g_{21}00-g_{24}-g_{25}00-g_{27}-g_{28}00-g_{30}00-g_{32}0000000
(1, 0, 1, 0, 0, 0)1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}g_{7}00g_{-31}0g_{-28}00g_{-23}0-g_{-20}00-g_{-16}0-g_{-14}000-g_{-10}-g_{-8}0000-g_{-4}0000h_{3}+h_{1}000g_{1}0-g_{3}-g_{7}0-g_{7}g_{7}00000g_{12}000g_{17}0g_{18}000g_{22}0g_{24}000g_{27}00-g_{29}0000-g_{32}00-g_{34}00000
(0, 1, 0, 1, 0, 0)-e_{1}-e_{3}g_{8}-g_{-34}00000g_{-24}0g_{-21}0g_{-18}00g_{-15}00-g_{-11}00g_{-7}00-g_{-5}g_{-3}0000h_{4}+h_{2}000g_{2}0-g_{4}00-g_{8}g_{8}-g_{8}g_{8}000-g_{13}0g_{14}0-g_{17}000g_{20}000-g_{23}00-g_{26}00-g_{28}00-g_{30}000000000g_{36}00
(0, 0, 1, 1, 0, 0)e_{1}-e_{3}g_{9}0g_{-33}00-g_{-27}00-g_{-22}g_{-20}0000g_{-14}0-g_{-11}00000-g_{-5}0g_{-2}g_{-1}00h_{4}+h_{3}0000g_{3}-g_{4}00g_{9}g_{9}-g_{9}-g_{9}g_{9}0-g_{12}-g_{13}00g_{15}00000g_{21}00-g_{23}00000-g_{28}0g_{29}0000g_{32}00000-g_{35}000
(0, 0, 0, 1, 1, 0)e_{2}-e_{4}g_{10}0-g_{-32}0g_{-27}0g_{-25}0000g_{-17}00g_{-13}0000g_{-7}0-g_{-6}g_{-3}g_{-2}000h_{5}+h_{4}0000g_{4}-g_{5}0000g_{10}g_{10}-g_{10}-g_{10}g_{10}0-g_{14}-g_{15}00g_{16}-g_{18}00000-g_{23}000-g_{26}0000000-g_{31}0-g_{33}0000g_{35}0000
(0, 0, 0, 0, 1, 1)e_{3}-e_{5}g_{11}00g_{-29}g_{-26}0g_{-23}000g_{-17}0g_{-13}g_{-12}00g_{-9}g_{-8}000g_{-4}0000h_{6}+h_{5}0000g_{5}-g_{6}0000000g_{11}-g_{11}-g_{11}000-g_{16}000-g_{20}-g_{21}00-g_{24}-g_{25}000-g_{27}00000-g_{31}00-g_{33}00-g_{34}0000000
(1, 0, 1, 1, 0, 0)1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}g_{12}0-g_{-31}00g_{-25}0g_{-20}g_{-19}00g_{-14}0-g_{-11}00000-g_{-5}g_{-2}0000h_{4}+h_{3}+h_{1}00g_{1}0-g_{4}00g_{7}00-g_{9}-g_{12}g_{12}0-g_{12}g_{12}00-g_{17}00g_{18}00000g_{24}00-g_{26}0000-g_{29}-g_{30}0000-g_{32}00000g_{35}00000
(0, 1, 1, 1, 0, 0)-e_{2}-e_{3}g_{13}g_{-33}000g_{-24}00g_{-18}-g_{-16}00-g_{-11}0-g_{-10}000-g_{-5}0g_{-1}000h_{4}+h_{3}+h_{2}000g_{2}g_{3}0000-g_{8}-g_{9}0g_{13}-g_{13}-g_{13}0g_{13}0-g_{17}000g_{19}0000g_{23}g_{25}0000g_{28}0-g_{29}00000-g_{32}00000000-g_{36}000
(0, 1, 0, 1, 1, 0)-e_{1}-e_{4}g_{14}-g_{-32}00-g_{-24}0-g_{-21}0000-g_{-12}00-g_{-9}g_{-7}0-g_{-6}g_{-3}0000h_{5}+h_{4}+h_{2}000g_{2}0-g_{5}00g_{8}00-g_{10}00-g_{14}g_{14}0-g_{14}g_{14}00-g_{19}00g_{20}-g_{22}0g_{23}00g_{26}00000000g_{31}00g_{33}0000000g_{36}0000
(0, 0, 1, 1, 1, 0)e_{1}-e_{4}g_{15}0g_{-30}g_{-27}00-g_{-20}0g_{-17}00000-g_{-8}0-g_{-6}0g_{-2}g_{-1}00h_{5}+h_{4}+h_{3}0000g_{3}-g_{5}000g_{9}0-g_{10}00g_{15}g_{15}-g_{15}0-g_{15}g_{15}-g_{18}-g_{19}000g_{21}0g_{23}00000000-g_{29}00g_{31}000000-g_{34}00-g_{35}000000
(0, 0, 0, 1, 1, 1)e_{2}-e_{5}g_{16}0g_{-29}0-g_{-22}0-g_{-19}g_{-17}0g_{-13}000g_{-7}00g_{-3}g_{-2}000h_{6}+h_{5}+h_{4}0000g_{4}-g_{6}000g_{10}0-g_{11}0000g_{16}g_{16}-g_{16}0-g_{16}0-g_{20}-g_{21}000-g_{24}00000-g_{28}000-g_{30}0g_{31}00g_{33}000000-g_{35}0000000
(1, 1, 1, 1, 0, 0)-1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}g_{17}-g_{-31}000-g_{-21}0-g_{-16}-g_{-15}0-g_{-11}-g_{-10}000-g_{-5}0000h_{4}+h_{3}+h_{2}+h_{1}000g_{1}g_{2}000g_{7}-g_{8}0000-g_{12}-g_{13}-g_{17}-g_{17}00g_{17}00000g_{22}0000g_{26}g_{27}000g_{29}g_{30}0000g_{32}000000000g_{36}00000
(1, 0, 1, 1, 1, 0)1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}g_{18}0-g_{-28}-g_{-25}-g_{-20}000-g_{-13}00-g_{-8}0-g_{-6}0g_{-2}000h_{5}+h_{4}+h_{3}+h_{1}00g_{1}00-g_{5}0g_{7}00-g_{10}0g_{12}000-g_{15}-g_{18}g_{18}00-g_{18}g_{18}0-g_{22}000g_{24}0g_{26}0000g_{29}000000g_{33}0000g_{34}00g_{35}00000000
(0, 1, 1, 1, 1, 0)-e_{2}-e_{4}g_{19}g_{-30}0-g_{-24}00g_{-16}0-g_{-12}000-g_{-6}0g_{-4}g_{-1}00h_{5}+h_{4}+h_{3}+h_{2}000g_{2}g_{3}-g_{5}0000000g_{13}0-g_{14}-g_{15}0g_{19}-g_{19}-g_{19}g_{19}-g_{19}g_{19}-g_{22}00-g_{23}0g_{25}00000g_{29}000-g_{31}0000000g_{34}00000-g_{36}000000
(0, 1, 0, 1, 1, 1)-e_{1}-e_{5}g_{20}g_{-29}00g_{-18}0g_{-15}-g_{-12}0-g_{-9}g_{-7}0g_{-3}0000h_{6}+h_{5}+h_{4}+h_{2}000g_{2}0-g_{6}00g_{8}00-g_{11}0g_{14}000-g_{16}00-g_{20}g_{20}00-g_{20}00-g_{25}000-g_{27}0g_{28}00g_{30}00-g_{31}00-g_{33}0000000000-g_{36}0000000
(0, 0, 1, 1, 1, 1)e_{1}-e_{5}g_{21}0-g_{-26}-g_{-22}0g_{-17}g_{-14}00-g_{-8}00g_{-2}g_{-1}00h_{6}+h_{5}+h_{4}+h_{3}0000g_{3}-g_{6}000g_{9}0-g_{11}00g_{15}00-g_{16}00g_{21}g_{21}-g_{21}00-g_{21}-g_{24}-g_{25}00000g_{28}00000-g_{31}00-g_{32}0000g_{34}000g_{35}0000000000
(1, 1, 1, 1, 1, 0)-1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}g_{22}-g_{-28}0g_{-21}g_{-16}000g_{-9}0-g_{-6}g_{-4}000h_{5}+h_{4}+h_{3}+h_{2}+h_{1}00g_{1}g_{2}-g_{5}00g_{7}000000-g_{14}0g_{17}00-g_{18}-g_{19}-g_{22}-g_{22}0g_{22}-g_{22}g_{22}000-g_{26}0g_{27}00-g_{29}000000-g_{33}0000-g_{34}000000g_{36}00000000
(0, 1, 1, 2, 1, 0)-e_{3}-e_{4}g_{23}-g_{-27}-g_{-24}000-g_{-11}0-g_{-7}-g_{-6}0g_{-1}00h_{5}+2h_{4}+h_{3}+h_{2}000g_{4}000-g_{8}-g_{9}-g_{10}00g_{13}g_{14}g_{15}000-g_{19}000g_{23}00-g_{23}0g_{23}-g_{26}0000g_{28}g_{29}000g_{31}000000000000g_{35}00g_{36}000000000
(1, 0, 1, 1, 1, 1)1/2e_{1}+1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}g_{24}0g_{-23}g_{-19}g_{-14}-g_{-13}0-g_{-8}00g_{-2}00h_{6}+h_{5}+h_{4}+h_{3}+h_{1}00g_{1}00-g_{6}0g_{7}000-g_{11}g_{12}000-g_{16}g_{18}0000-g_{21}-g_{24}g_{24}000-g_{24}0-g_{27}00000g_{30}0000g_{32}-g_{33}0000-g_{34}000-g_{35}0000000000000
(0, 1, 1, 1, 1, 1)-e_{2}-e_{5}g_{25}-g_{-26}0g_{-18}0-g_{-12}-g_{-10}00g_{-4}g_{-1}0h_{6}+h_{5}+h_{4}+h_{3}+h_{2}000g_{2}g_{3}-g_{6}00000-g_{11}0g_{13}0000g_{19}00-g_{20}-g_{21}0g_{25}-g_{25}-g_{25}g_{25}0-g_{25}-g_{27}00-g_{28}00000g_{31}0g_{32}00000-g_{34}0000000g_{36}0000000000
(1, 1, 1, 2, 1, 0)-1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}g_{26}g_{-25}g_{-21}0-g_{-11}00-g_{-6}g_{-3}00h_{5}+2h_{4}+h_{3}+h_{2}+h_{1}00g_{1}g_{4}000-g_{8}-g_{10}00-g_{12}0g_{14}0g_{17}0g_{18}000-g_{22}00-g_{23}-g_{26}0g_{26}-g_{26}0g_{26}00-g_{29}00g_{30}0000g_{33}000000000-g_{35}000-g_{36}00000000000
(1, 1, 1, 1, 1, 1)-1/2e_{1}-1/2e_{2}+1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}g_{27}g_{-23}0-g_{-15}-g_{-10}g_{-9}0g_{-4}00h_{6}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}0g_{1}g_{2}0-g_{6}0g_{7}00-g_{11}00000g_{17}000-g_{20}g_{22}000-g_{24}-g_{25}-g_{27}-g_{27}0g_{27}0-g_{27}000-g_{30}0000-g_{32}g_{33}0000g_{34}0000000-g_{36}0000000000000
(0, 1, 1, 2, 1, 1)-e_{3}-e_{5}g_{28}g_{-22}g_{-18}00-g_{-7}g_{-5}g_{-1}0h_{6}+h_{5}+2h_{4}+h_{3}+h_{2}00g_{4}0-g_{6}0-g_{8}-g_{9}000g_{13}00-g_{16}000g_{20}g_{21}0g_{23}0-g_{25}000g_{28}00-g_{28}g_{28}-g_{28}-g_{30}000-g_{31}0g_{32}0000000000-g_{35}000-g_{36}00000000000000
(1, 1, 2, 2, 1, 0)1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}+1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}g_{29}-g_{-20}-g_{-16}-g_{-11}0-g_{-6}00h_{5}+2h_{4}+2h_{3}+h_{2}+h_{1}00g_{3}00-g_{7}g_{9}00-g_{12}-g_{13}-g_{15}0g_{17}0g_{18}g_{19}00-g_{22}0g_{23}000-g_{26}0000-g_{29}00g_{29}00000g_{32}0000g_{34}0000g_{35}000g_{36}0000000000000000
(1, 1, 1, 2, 1, 1)-1/2e_{1}+1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}g_{30}-g_{-19}-g_{-15}0g_{-5}g_{-3}0h_{6}+h_{5}+2h_{4}+h_{3}+h_{2}+h_{1}0g_{1}g_{4}-g_{6}0-g_{8}000-g_{12}00-g_{16}g_{17}000g_{20}000g_{24}0g_{26}0-g_{27}00-g_{28}-g_{30}0g_{30}-g_{30}g_{30}-g_{30}00-g_{32}0-g_{33}000000000g_{35}000g_{36}00000000000000000
(0, 1, 1, 2, 2, 1)-e_{4}-e_{5}g_{31}g_{-17}g_{-12}-g_{-7}g_{-1}0h_{6}+2h_{5}+2h_{4}+h_{3}+h_{2}00g_{5}00-g_{10}0-g_{11}0g_{14}g_{15}g_{16}00-g_{19}-g_{20}-g_{21}00g_{23}g_{25}0000-g_{28}0000g_{31}000-g_{31}0-g_{33}00000g_{34}0000-g_{35}0000-g_{36}0000000000000000000
(1, 1, 2, 2, 1, 1)1/2e_{1}-1/2e_{2}-1/2e_{3}+1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}g_{32}g_{-14}g_{-10}g_{-5}0h_{6}+h_{5}+2h_{4}+2h_{3}+h_{2}+h_{1}0g_{3}-g_{6}-g_{7}g_{9}0-g_{12}-g_{13}00g_{17}000-g_{21}000g_{24}g_{25}00-g_{27}0g_{28}g_{29}00-g_{30}0000-g_{32}0g_{32}-g_{32}0000-g_{34}0000-g_{35}000-g_{36}0000000000000000000000
(1, 1, 1, 2, 2, 1)-1/2e_{1}+1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}g_{33}-g_{-13}-g_{-9}g_{-3}h_{6}+2h_{5}+2h_{4}+h_{3}+h_{2}+h_{1}0g_{1}g_{5}00-g_{10}-g_{11}0g_{14}0g_{16}0g_{18}0-g_{20}0-g_{22}0-g_{24}00g_{26}g_{27}0000-g_{30}000-g_{31}-g_{33}0g_{33}0-g_{33}000-g_{34}00000g_{35}000g_{36}00000000000000000000000
(1, 1, 2, 2, 2, 1)1/2e_{1}-1/2e_{2}+1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}g_{34}g_{-8}g_{-4}h_{6}+2h_{5}+2h_{4}+2h_{3}+h_{2}+h_{1}g_{3}g_{5}-g_{7}0-g_{11}0-g_{15}0g_{18}g_{19}0g_{21}-g_{22}0-g_{24}-g_{25}00g_{27}000g_{29}000g_{31}0-g_{32}0-g_{33}0000-g_{34}g_{34}-g_{34}0000-g_{35}000-g_{36}0000000000000000000000000000
(1, 1, 2, 3, 2, 1)1/2e_{1}+1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}g_{35}g_{-2}h_{6}+2h_{5}+3h_{4}+2h_{3}+h_{2}+h_{1}g_{4}-g_{9}g_{10}g_{12}-g_{15}-g_{16}g_{18}0g_{21}0g_{23}-g_{24}0-g_{26}00-g_{28}0g_{29}g_{30}00-g_{31}0-g_{32}g_{33}0000-g_{34}0000g_{35}0-g_{35}000-g_{36}0000000000000000000000000000000000
(1, 2, 2, 3, 2, 1)-1/2e_{1}-1/2e_{2}-1/2e_{3}-1/2e_{4}-1/2e_{5}+1/2e_{6}+1/2e_{7}-1/2e_{8}g_{36}h_{6}+2h_{5}+3h_{4}+2h_{3}+2h_{2}+h_{1}g_{2}g_{8}-g_{13}g_{14}g_{17}-g_{19}-g_{20}g_{22}g_{23}g_{25}-g_{26}0-g_{27}-g_{28}0g_{29}g_{30}0-g_{31}00-g_{32}g_{33}0000-g_{34}00000-g_{35}00-g_{36}0000000000000000000000000000000000000000
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:
\(\displaystyle \begin{pmatrix}2 & 0 & -1 & 0 & 0 & 0\\ 0 & 2 & 0 & -1 & 0 & 0\\ -1 & 0 & 2 & -1 & 0 & 0\\ 0 & -1 & -1 & 2 & -1 & 0\\ 0 & 0 & 0 & -1 & 2 & -1\\ 0 & 0 & 0 & 0 & -1 & 2\\ \end{pmatrix}\)
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:
\(\displaystyle \begin{pmatrix}2 & 0 & -1 & 0 & 0 & 0\\ 0 & 2 & 0 & -1 & 0 & 0\\ -1 & 0 & 2 & -1 & 0 & 0\\ 0 & -1 & -1 & 2 & -1 & 0\\ 0 & 0 & 0 & -1 & 2 & -1\\ 0 & 0 & 0 & 0 & -1 & 2\\ \end{pmatrix}\)
The determinant of the symmetric Cartan matrix is: 3
Half sum of positive roots: (8, 11, 15, 21, 15, 8)= \(\displaystyle -\varepsilon_{2}-2\varepsilon_{3}-3\varepsilon_{4}-4\varepsilon_{5}+4\varepsilon_{6}+4\varepsilon_{7}-4\varepsilon_{8}\)
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):
(4/3, 1, 5/3, 2, 4/3, 2/3) = \(\displaystyle 2/3\varepsilon_{6}+2/3\varepsilon_{7}-2/3\varepsilon_{8}\)
(1, 2, 2, 3, 2, 1) = \(\displaystyle -1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}\)
(5/3, 2, 10/3, 4, 8/3, 4/3) = \(\displaystyle 1/2\varepsilon_{1}-1/2\varepsilon_{2}-1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}+5/6\varepsilon_{6}+5/6\varepsilon_{7}-5/6\varepsilon_{8}\)
(2, 3, 4, 6, 4, 2) = \(\displaystyle -\varepsilon_{3}-\varepsilon_{4}-\varepsilon_{5}+\varepsilon_{6}+\varepsilon_{7}-\varepsilon_{8}\)
(4/3, 2, 8/3, 4, 10/3, 5/3) = \(\displaystyle -\varepsilon_{4}-\varepsilon_{5}+2/3\varepsilon_{6}+2/3\varepsilon_{7}-2/3\varepsilon_{8}\)
(2/3, 1, 4/3, 2, 5/3, 4/3) = \(\displaystyle -\varepsilon_{5}+1/3\varepsilon_{6}+1/3\varepsilon_{7}-1/3\varepsilon_{8}\)
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) = \(\displaystyle -1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}-1/2\varepsilon_{8}\)
(0, 1, 0, 0, 0, 0) = \(\displaystyle -\varepsilon_{1}-\varepsilon_{2}\)
(0, 0, 1, 0, 0, 0) = \(\displaystyle \varepsilon_{1}-\varepsilon_{2}\)
(0, 0, 0, 1, 0, 0) = \(\displaystyle \varepsilon_{2}-\varepsilon_{3}\)
(0, 0, 0, 0, 1, 0) = \(\displaystyle \varepsilon_{3}-\varepsilon_{4}\)
(0, 0, 0, 0, 0, 1) = \(\displaystyle \varepsilon_{4}-\varepsilon_{5}\)