so(14), type \(D^{1}_7\)
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Page generated by the calculator project.
Lie algebra type: D^{1}_7.
Weyl group size: 322560.
A drawing of the root system in its corresponding Coxeter plane. Computations were carried out as explained by John Stembridge.
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The grey lines are the edges of the Weyl chamber.
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The root system has 84 elements.
Simple basis coordinatesEpsilon coordinatesReflection w.r.t. root
(-1, -2, -2, -2, -2, -1, -1)-e_{1}-e_{2}\(s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}s_{1}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -2, -2, -2, -1, -1)-e_{1}-e_{3}\(s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}s_{1}s_{6}s_{5}s_{4}s_{3}\)
(0, -1, -2, -2, -2, -1, -1)-e_{2}-e_{3}\(s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}s_{6}s_{5}s_{4}s_{3}\)
(-1, -1, -1, -2, -2, -1, -1)-e_{1}-e_{4}\(s_{1}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}s_{1}s_{6}s_{5}s_{4}\)
(0, -1, -1, -2, -2, -1, -1)-e_{2}-e_{4}\(s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}s_{6}s_{5}s_{4}\)
(-1, -1, -1, -1, -2, -1, -1)-e_{1}-e_{5}\(s_{1}s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}s_{1}s_{6}s_{5}\)
(0, 0, -1, -2, -2, -1, -1)-e_{3}-e_{4}\(s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{6}s_{5}s_{4}\)
(0, -1, -1, -1, -2, -1, -1)-e_{2}-e_{5}\(s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}s_{6}s_{5}\)
(-1, -1, -1, -1, -1, -1, -1)-e_{1}-e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}s_{1}s_{6}\)
(0, 0, -1, -1, -2, -1, -1)-e_{3}-e_{5}\(s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{6}s_{5}\)
(0, -1, -1, -1, -1, -1, -1)-e_{2}-e_{6}\(s_{2}s_{3}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}s_{6}\)
(-1, -1, -1, -1, -1, 0, -1)-e_{1}-e_{7}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(-1, -1, -1, -1, -1, -1, 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, -1, -2, -1, -1)-e_{4}-e_{5}\(s_{5}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{6}s_{5}\)
(0, 0, -1, -1, -1, -1, -1)-e_{3}-e_{6}\(s_{3}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{6}\)
(0, -1, -1, -1, -1, 0, -1)-e_{2}-e_{7}\(s_{2}s_{3}s_{4}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}\)
(0, -1, -1, -1, -1, -1, 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)-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, -1, -1, -1, -1)-e_{4}-e_{6}\(s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{6}\)
(0, 0, -1, -1, -1, 0, -1)-e_{3}-e_{7}\(s_{3}s_{4}s_{5}s_{7}s_{5}s_{4}s_{3}\)
(0, 0, -1, -1, -1, -1, 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)-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)-e_{1}+e_{5}\(s_{1}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, -1, -1, -1)-e_{5}-e_{6}\(s_{6}s_{5}s_{7}s_{5}s_{6}\)
(0, 0, 0, -1, -1, 0, -1)-e_{4}-e_{7}\(s_{4}s_{5}s_{7}s_{5}s_{4}\)
(0, 0, 0, -1, -1, -1, 0)-e_{4}+e_{7}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(0, 0, -1, -1, -1, 0, 0)-e_{3}+e_{6}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, -1, -1, -1, 0, 0, 0)-e_{2}+e_{5}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(-1, -1, -1, 0, 0, 0, 0)-e_{1}+e_{4}\(s_{1}s_{2}s_{3}s_{2}s_{1}\)
(0, 0, 0, 0, -1, 0, -1)-e_{5}-e_{7}\(s_{5}s_{7}s_{5}\)
(0, 0, 0, 0, -1, -1, 0)-e_{5}+e_{7}\(s_{5}s_{6}s_{5}\)
(0, 0, 0, -1, -1, 0, 0)-e_{4}+e_{6}\(s_{4}s_{5}s_{4}\)
(0, 0, -1, -1, 0, 0, 0)-e_{3}+e_{5}\(s_{3}s_{4}s_{3}\)
(0, -1, -1, 0, 0, 0, 0)-e_{2}+e_{4}\(s_{2}s_{3}s_{2}\)
(-1, -1, 0, 0, 0, 0, 0)-e_{1}+e_{3}\(s_{1}s_{2}s_{1}\)
(0, 0, 0, 0, 0, 0, -1)-e_{6}-e_{7}\(s_{7}\)
(0, 0, 0, 0, 0, -1, 0)-e_{6}+e_{7}\(s_{6}\)
(0, 0, 0, 0, -1, 0, 0)-e_{5}+e_{6}\(s_{5}\)
(0, 0, 0, -1, 0, 0, 0)-e_{4}+e_{5}\(s_{4}\)
(0, 0, -1, 0, 0, 0, 0)-e_{3}+e_{4}\(s_{3}\)
(0, -1, 0, 0, 0, 0, 0)-e_{2}+e_{3}\(s_{2}\)
(-1, 0, 0, 0, 0, 0, 0)-e_{1}+e_{2}\(s_{1}\)
(1, 0, 0, 0, 0, 0, 0)e_{1}-e_{2}\(s_{1}\)
(0, 1, 0, 0, 0, 0, 0)e_{2}-e_{3}\(s_{2}\)
(0, 0, 1, 0, 0, 0, 0)e_{3}-e_{4}\(s_{3}\)
(0, 0, 0, 1, 0, 0, 0)e_{4}-e_{5}\(s_{4}\)
(0, 0, 0, 0, 1, 0, 0)e_{5}-e_{6}\(s_{5}\)
(0, 0, 0, 0, 0, 1, 0)e_{6}-e_{7}\(s_{6}\)
(0, 0, 0, 0, 0, 0, 1)e_{6}+e_{7}\(s_{7}\)
(1, 1, 0, 0, 0, 0, 0)e_{1}-e_{3}\(s_{1}s_{2}s_{1}\)
(0, 1, 1, 0, 0, 0, 0)e_{2}-e_{4}\(s_{2}s_{3}s_{2}\)
(0, 0, 1, 1, 0, 0, 0)e_{3}-e_{5}\(s_{3}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 0, 0)e_{4}-e_{6}\(s_{4}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 0)e_{5}-e_{7}\(s_{5}s_{6}s_{5}\)
(0, 0, 0, 0, 1, 0, 1)e_{5}+e_{7}\(s_{5}s_{7}s_{5}\)
(1, 1, 1, 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)e_{2}-e_{5}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 0, 0)e_{3}-e_{6}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1, 0)e_{4}-e_{7}\(s_{4}s_{5}s_{6}s_{5}s_{4}\)
(0, 0, 0, 1, 1, 0, 1)e_{4}+e_{7}\(s_{4}s_{5}s_{7}s_{5}s_{4}\)
(0, 0, 0, 0, 1, 1, 1)e_{5}+e_{6}\(s_{6}s_{5}s_{7}s_{5}s_{6}\)
(1, 1, 1, 1, 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)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)e_{3}-e_{7}\(s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}\)
(0, 0, 1, 1, 1, 0, 1)e_{3}+e_{7}\(s_{3}s_{4}s_{5}s_{7}s_{5}s_{4}s_{3}\)
(0, 0, 0, 1, 1, 1, 1)e_{4}+e_{6}\(s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{6}\)
(1, 1, 1, 1, 1, 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)e_{2}-e_{7}\(s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}\)
(0, 1, 1, 1, 1, 0, 1)e_{2}+e_{7}\(s_{2}s_{3}s_{4}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1, 1, 1)e_{3}+e_{6}\(s_{3}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{6}\)
(0, 0, 0, 1, 2, 1, 1)e_{4}+e_{5}\(s_{5}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{6}s_{5}\)
(1, 1, 1, 1, 1, 1, 0)e_{1}-e_{7}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(1, 1, 1, 1, 1, 0, 1)e_{1}+e_{7}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 1, 1, 1)e_{2}+e_{6}\(s_{2}s_{3}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}s_{6}\)
(0, 0, 1, 1, 2, 1, 1)e_{3}+e_{5}\(s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{6}s_{5}\)
(1, 1, 1, 1, 1, 1, 1)e_{1}+e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}s_{1}s_{6}\)
(0, 1, 1, 1, 2, 1, 1)e_{2}+e_{5}\(s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}s_{6}s_{5}\)
(0, 0, 1, 2, 2, 1, 1)e_{3}+e_{4}\(s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{6}s_{5}s_{4}\)
(1, 1, 1, 1, 2, 1, 1)e_{1}+e_{5}\(s_{1}s_{2}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}s_{1}s_{6}s_{5}\)
(0, 1, 1, 2, 2, 1, 1)e_{2}+e_{4}\(s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}s_{6}s_{5}s_{4}\)
(1, 1, 1, 2, 2, 1, 1)e_{1}+e_{4}\(s_{1}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}s_{1}s_{6}s_{5}s_{4}\)
(0, 1, 2, 2, 2, 1, 1)e_{2}+e_{3}\(s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}s_{6}s_{5}s_{4}s_{3}\)
(1, 1, 2, 2, 2, 1, 1)e_{1}+e_{3}\(s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}s_{1}s_{6}s_{5}s_{4}s_{3}\)
(1, 2, 2, 2, 2, 1, 1)e_{1}+e_{2}\(s_{2}s_{1}s_{3}s_{2}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}s_{7}s_{5}s_{4}s_{3}s_{2}s_{1}s_{6}s_{5}s_{4}s_{3}s_{2}\)
Comma delimited list of roots: (-1, -2, -2, -2, -2, -1, -1), (-1, -1, -2, -2, -2, -1, -1), (0, -1, -2, -2, -2, -1, -1), (-1, -1, -1, -2, -2, -1, -1), (0, -1, -1, -2, -2, -1, -1), (-1, -1, -1, -1, -2, -1, -1), (0, 0, -1, -2, -2, -1, -1), (0, -1, -1, -1, -2, -1, -1), (-1, -1, -1, -1, -1, -1, -1), (0, 0, -1, -1, -2, -1, -1), (0, -1, -1, -1, -1, -1, -1), (-1, -1, -1, -1, -1, 0, -1), (-1, -1, -1, -1, -1, -1, 0), (0, 0, 0, -1, -2, -1, -1), (0, 0, -1, -1, -1, -1, -1), (0, -1, -1, -1, -1, 0, -1), (0, -1, -1, -1, -1, -1, 0), (-1, -1, -1, -1, -1, 0, 0), (0, 0, 0, -1, -1, -1, -1), (0, 0, -1, -1, -1, 0, -1), (0, 0, -1, -1, -1, -1, 0), (0, -1, -1, -1, -1, 0, 0), (-1, -1, -1, -1, 0, 0, 0), (0, 0, 0, 0, -1, -1, -1), (0, 0, 0, -1, -1, 0, -1), (0, 0, 0, -1, -1, -1, 0), (0, 0, -1, -1, -1, 0, 0), (0, -1, -1, -1, 0, 0, 0), (-1, -1, -1, 0, 0, 0, 0), (0, 0, 0, 0, -1, 0, -1), (0, 0, 0, 0, -1, -1, 0), (0, 0, 0, -1, -1, 0, 0), (0, 0, -1, -1, 0, 0, 0), (0, -1, -1, 0, 0, 0, 0), (-1, -1, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, -1), (0, 0, 0, 0, 0, -1, 0), (0, 0, 0, 0, -1, 0, 0), (0, 0, 0, -1, 0, 0, 0), (0, 0, -1, 0, 0, 0, 0), (0, -1, 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, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1), (1, 1, 0, 0, 0, 0, 0), (0, 1, 1, 0, 0, 0, 0), (0, 0, 1, 1, 0, 0, 0), (0, 0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 1, 1, 0), (0, 0, 0, 0, 1, 0, 1), (1, 1, 1, 0, 0, 0, 0), (0, 1, 1, 1, 0, 0, 0), (0, 0, 1, 1, 1, 0, 0), (0, 0, 0, 1, 1, 1, 0), (0, 0, 0, 1, 1, 0, 1), (0, 0, 0, 0, 1, 1, 1), (1, 1, 1, 1, 0, 0, 0), (0, 1, 1, 1, 1, 0, 0), (0, 0, 1, 1, 1, 1, 0), (0, 0, 1, 1, 1, 0, 1), (0, 0, 0, 1, 1, 1, 1), (1, 1, 1, 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, 0, 1, 2, 1, 1), (1, 1, 1, 1, 1, 1, 0), (1, 1, 1, 1, 1, 0, 1), (0, 1, 1, 1, 1, 1, 1), (0, 0, 1, 1, 2, 1, 1), (1, 1, 1, 1, 1, 1, 1), (0, 1, 1, 1, 2, 1, 1), (0, 0, 1, 2, 2, 1, 1), (1, 1, 1, 1, 2, 1, 1), (0, 1, 1, 2, 2, 1, 1), (1, 1, 1, 2, 2, 1, 1), (0, 1, 2, 2, 2, 1, 1), (1, 1, 2, 2, 2, 1, 1), (1, 2, 2, 2, 2, 1, 1) The resulting Lie bracket pairing table follows.
Type D^{1}_7.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_{-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}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}
(-1, -2, -2, -2, -2, -1, -1)-e_{1}-e_{2}g_{-42}0000000000000000000000000000000000000000000g_{-42}000000g_{-41}00000-g_{-40}g_{-39}0000-g_{-38}g_{-37}0000-g_{-35}g_{-34}000-g_{-32}g_{-31}g_{-30}00-g_{-27}-g_{-26}g_{-25}0-g_{-21}g_{-20}0-g_{-15}g_{-14}-g_{-9}g_{-8}-g_{-2}-h_{7}-h_{6}-2h_{5}-2h_{4}-2h_{3}-2h_{2}-h_{1}
(-1, -1, -2, -2, -2, -1, -1)-e_{1}-e_{3}g_{-41}0000000000000000000000000000000000000000g_{-42}0g_{-41}-g_{-41}g_{-41}0000g_{-40}0g_{-39}000000g_{-37}000-g_{-36}0g_{-34}000-g_{-33}0g_{-31}g_{-30}0-g_{-28}00g_{-25}0-g_{-23}-g_{-22}0g_{-20}-g_{-16}0g_{-14}-g_{-10}0-g_{-3}-g_{-1}-h_{7}-h_{6}-2h_{5}-2h_{4}-2h_{3}-h_{2}-h_{1}-g_{2}
(0, -1, -2, -2, -2, -1, -1)-e_{2}-e_{3}g_{-40}0000000000000000000000000000000000-g_{-42}000000g_{-41}-g_{-40}0g_{-40}000000g_{-38}00000-g_{-36}g_{-35}0000-g_{-33}g_{-32}0000-g_{-28}g_{-27}g_{-26}00-g_{-23}-g_{-22}g_{-21}000-g_{-16}g_{-15}0-g_{-10}g_{-9}0-g_{-3}0-h_{7}-h_{6}-2h_{5}-2h_{4}-2h_{3}-h_{2}-g_{1}g_{8}
(-1, -1, -1, -2, -2, -1, -1)-e_{1}-e_{4}g_{-39}000000000000000000000000000000000g_{-42}00000g_{-41}00g_{-39}0-g_{-39}g_{-39}000g_{-38}00g_{-37}000g_{-36}00g_{-34}00000g_{-31}g_{-30}0-g_{-29}000g_{-25}-g_{-24}000g_{-20}-g_{-18}-g_{-17}00-g_{-11}0-g_{-8}-g_{-4}-g_{-1}-h_{7}-h_{6}-2h_{5}-2h_{4}-h_{3}-h_{2}-h_{1}0-g_{3}-g_{9}
(0, -1, -1, -2, -2, -1, -1)-e_{2}-e_{4}g_{-38}0000000000000000000000000000-g_{-42}0000000000g_{-40}0g_{-39}-g_{-38}g_{-38}-g_{-38}g_{-38}0000g_{-36}0g_{-35}000000g_{-32}000-g_{-29}0g_{-27}g_{-26}00-g_{-24}00g_{-21}0-g_{-18}-g_{-17}0g_{-15}00-g_{-11}00-g_{-4}-g_{-2}0-h_{7}-h_{6}-2h_{5}-2h_{4}-h_{3}-h_{2}-g_{1}-g_{3}0g_{14}
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(1, 0, 0, 0, 0, 0, 0)e_{1}-e_{2}g_{1}0-g_{-40}0-g_{-38}0-g_{-35}00-g_{-32}00-g_{-27}-g_{-26}0000-g_{-21}0000-g_{-15}00000-g_{-9}00000-g_{-2}000000h_{1}-2g_{1}g_{1}000000g_{8}000000g_{14}00000g_{20}00000g_{25}0000g_{30}g_{31}0000g_{34}00g_{37}00g_{39}0g_{41}00
(0, 1, 0, 0, 0, 0, 0)e_{2}-e_{3}g_{2}-g_{-41}000-g_{-36}00-g_{-33}00-g_{-28}0000-g_{-23}-g_{-22}0000-g_{-16}00000-g_{-10}00000-g_{-3}g_{-1}00000h_{2}0g_{2}-2g_{2}g_{2}0000-g_{8}0g_{9}000000g_{15}00000g_{21}00000g_{26}g_{27}0000g_{32}0000g_{35}00g_{38}0000g_{42}0
(0, 0, 1, 0, 0, 0, 0)e_{3}-e_{4}g_{3}0-g_{-39}-g_{-38}000000-g_{-29}0000-g_{-24}0000-g_{-18}-g_{-17}00000-g_{-11}0g_{-8}000-g_{-4}g_{-2}00000h_{3}000g_{3}-2g_{3}g_{3}0000-g_{9}0g_{10}000-g_{14}00g_{16}00000g_{22}g_{23}00000g_{28}0000g_{33}00000000g_{40}g_{41}000
(0, 0, 0, 1, 0, 0, 0)e_{4}-e_{5}g_{4}000-g_{-37}-g_{-35}0-g_{-33}00000000000-g_{-19}000g_{-14}0-g_{-13}-g_{-12}0g_{-9}000-g_{-5}g_{-3}00000h_{4}00000g_{4}-2g_{4}g_{4}0000-g_{10}0g_{11}000-g_{15}00g_{17}g_{18}-g_{20}0000g_{24}0000000000000g_{36}0g_{38}0g_{39}00000
(0, 0, 0, 0, 1, 0, 0)e_{5}-e_{6}g_{5}00000-g_{-34}0-g_{-32}0-g_{-28}000-g_{-24}000g_{-20}000g_{-15}0000g_{-10}00-g_{-7}-g_{-6}g_{-4}00000h_{5}0000000g_{5}-2g_{5}g_{5}g_{5}000-g_{11}0g_{12}g_{13}00-g_{16}0000-g_{21}0000-g_{25}000g_{29}000g_{33}000g_{35}0g_{37}00000000
(0, 0, 0, 0, 0, 1, 0)e_{6}-e_{7}g_{6}00000000-g_{-31}0-g_{-27}0g_{-25}0-g_{-23}0g_{-21}0-g_{-18}0g_{-16}00-g_{-13}0g_{-11}0000g_{-5}00000h_{6}000000000g_{6}-2g_{6}00000-g_{12}00000-g_{17}0g_{19}00-g_{22}0g_{24}00-g_{26}0g_{28}0-g_{30}0g_{32}000g_{34}00000000000
(0, 0, 0, 0, 0, 0, 1)e_{6}+e_{7}g_{7}00000000-g_{-30}0-g_{-26}g_{-25}00-g_{-22}g_{-21}00-g_{-17}g_{-16}000-g_{-12}g_{-11}0000g_{-5}00000h_{7}0000000000g_{7}0-2g_{7}0000-g_{13}00000-g_{18}g_{19}000-g_{23}g_{24}000-g_{27}g_{28}00-g_{31}g_{32}000g_{34}000000000000
(1, 1, 0, 0, 0, 0, 0)e_{1}-e_{3}g_{8}g_{-40}00-g_{-36}0-g_{-33}00-g_{-28}00-g_{-23}-g_{-22}0000-g_{-16}0000-g_{-10}00000-g_{-3}00000h_{2}+h_{1}00000g_{1}-g_{2}-g_{8}-g_{8}g_{8}000000g_{14}000000g_{20}00000g_{25}00000g_{30}g_{31}0000g_{34}0000g_{37}00g_{39}000-g_{42}00
(0, 1, 1, 0, 0, 0, 0)e_{2}-e_{4}g_{9}-g_{-39}0g_{-36}0000-g_{-29}00-g_{-24}0000-g_{-18}-g_{-17}0000-g_{-11}00000-g_{-4}g_{-1}0000h_{3}+h_{2}00000g_{2}-g_{3}0g_{9}-g_{9}-g_{9}g_{9}000-g_{14}00g_{15}000000g_{21}00000g_{26}g_{27}00000g_{32}0000g_{35}000000-g_{40}00g_{42}000
(0, 0, 1, 1, 0, 0, 0)e_{3}-e_{5}g_{10}0-g_{-37}-g_{-35}000g_{-29}0000000-g_{-19}0000-g_{-13}-g_{-12}0g_{-8}000-g_{-5}g_{-2}0000h_{4}+h_{3}00000g_{3}-g_{4}000g_{10}-g_{10}-g_{10}g_{10}000-g_{15}00g_{16}00-g_{20}000g_{22}g_{23}00000g_{28}000000000-g_{36}00000g_{40}0g_{41}00000
(0, 0, 0, 1, 1, 0, 0)e_{4}-e_{6}g_{11}000-g_{-34}-g_{-32}0-g_{-28}000000g_{-19}000g_{-14}000g_{-9}00-g_{-7}-g_{-6}g_{-3}0000h_{5}+h_{4}00000g_{4}-g_{5}00000g_{11}-g_{11}-g_{11}g_{11}g_{11}00-g_{16}00g_{17}g_{18}0-g_{21}0000-g_{25}0000-g_{29}00000000g_{36}000g_{38}0g_{39}00000000
(0, 0, 0, 0, 1, 1, 0)e_{5}-e_{7}g_{12}00000-g_{-31}0-g_{-27}0-g_{-23}00g_{-20}-g_{-18}00g_{-15}000g_{-10}00g_{-7}0g_{-4}0000h_{6}+h_{5}00000g_{5}-g_{6}0000000g_{12}-g_{12}-g_{12}g_{12}000-g_{17}00-g_{19}00-g_{22}0000-g_{26}00g_{29}0-g_{30}00g_{33}000g_{35}000g_{37}00000000000
(0, 0, 0, 0, 1, 0, 1)e_{5}+e_{7}g_{13}00000-g_{-30}0-g_{-26}0-g_{-22}0g_{-20}0-g_{-17}0g_{-15}000g_{-10}000g_{-6}g_{-4}0000h_{7}+h_{5}00000g_{5}0-g_{7}0000000g_{13}-g_{13}g_{13}-g_{13}000-g_{18}0-g_{19}000-g_{23}0000-g_{27}0g_{29}00-g_{31}0g_{33}000g_{35}000g_{37}000000000000
(1, 1, 1, 0, 0, 0, 0)e_{1}-e_{4}g_{14}g_{-38}g_{-36}000-g_{-29}00-g_{-24}00-g_{-18}-g_{-17}0000-g_{-11}0000-g_{-4}00000h_{3}+h_{2}+h_{1}0000g_{1}-g_{3}0000g_{8}0-g_{9}-g_{14}0-g_{14}g_{14}000000g_{20}000000g_{25}00000g_{30}g_{31}00000g_{34}0000g_{37}000000-g_{41}0-g_{42}0000
(0, 1, 1, 1, 0, 0, 0)e_{2}-e_{5}g_{15}-g_{-37}0g_{-33}0g_{-29}00000-g_{-19}0000-g_{-13}-g_{-12}0000-g_{-5}g_{-1}0000h_{4}+h_{3}+h_{2}0000g_{2}-g_{4}0000g_{9}0-g_{10}0g_{15}-g_{15}0-g_{15}g_{15}00-g_{20}000g_{21}000000g_{26}g_{27}00000g_{32}000000000-g_{38}000-g_{40}000g_{42}00000
(0, 0, 1, 1, 1, 0, 0)e_{3}-e_{6}g_{16}0-g_{-34}-g_{-32}000g_{-24}00g_{-19}0000000g_{-8}0-g_{-7}-g_{-6}g_{-2}0000h_{5}+h_{4}+h_{3}0000g_{3}-g_{5}0000g_{10}0-g_{11}000g_{16}-g_{16}0-g_{16}g_{16}g_{16}0-g_{21}000g_{22}g_{23}-g_{25}0000000000-g_{33}0000-g_{36}0000000g_{40}0g_{41}00000000
(0, 0, 0, 1, 1, 1, 0)e_{4}-e_{7}g_{17}000-g_{-31}-g_{-27}0-g_{-23}00000g_{-14}g_{-13}00g_{-9}0g_{-7}0g_{-3}0000h_{6}+h_{5}+h_{4}0000g_{4}-g_{6}0000g_{11}0-g_{12}00000g_{17}-g_{17}0-g_{17}g_{17}00-g_{22}000-g_{24}0-g_{26}000-g_{29}-g_{30}00000000g_{36}000g_{38}000g_{39}00000000000
(0, 0, 0, 1, 1, 0, 1)e_{4}+e_{7}g_{18}000-g_{-30}-g_{-26}0-g_{-22}0000g_{-14}0g_{-12}0g_{-9}00g_{-6}g_{-3}0000h_{7}+h_{5}+h_{4}0000g_{4}0-g_{7}000g_{11}00-g_{13}00000g_{18}-g_{18}0g_{18}-g_{18}00-g_{23}00-g_{24}00-g_{27}00-g_{29}0-g_{31}0000000g_{36}000g_{38}000g_{39}000000000000
(0, 0, 0, 0, 1, 1, 1)e_{5}+e_{6}g_{19}00000-g_{-25}0-g_{-21}g_{-20}-g_{-16}g_{-15}00-g_{-11}g_{-10}000g_{-4}0000h_{7}+h_{6}+h_{5}00000g_{6}g_{7}0000-g_{12}-g_{13}00000000g_{19}0-g_{19}-g_{19}000-g_{24}00000-g_{28}g_{29}000-g_{32}g_{33}000-g_{34}g_{35}000g_{37}00000000000000000
(1, 1, 1, 1, 0, 0, 0)e_{1}-e_{5}g_{20}g_{-35}g_{-33}0g_{-29}0000-g_{-19}00-g_{-13}-g_{-12}0000-g_{-5}0000h_{4}+h_{3}+h_{2}+h_{1}0000g_{1}-g_{4}000g_{8}0-g_{10}000g_{14}00-g_{15}-g_{20}00-g_{20}g_{20}000000g_{25}000000g_{30}g_{31}00000g_{34}000000000-g_{39}000-g_{41}0-g_{42}0000000
(0, 1, 1, 1, 1, 0, 0)e_{2}-e_{6}g_{21}-g_{-34}0g_{-28}0g_{-24}00g_{-19}0000000-g_{-7}-g_{-6}g_{-1}000h_{5}+h_{4}+h_{3}+h_{2}0000g_{2}-g_{5}000g_{9}0-g_{11}000g_{15}00-g_{16}0g_{21}-g_{21}00-g_{21}g_{21}g_{21}-g_{25}0000g_{26}g_{27}00000000000-g_{35}0000-g_{38}000-g_{40}00000g_{42}00000000
(0, 0, 1, 1, 1, 1, 0)e_{3}-e_{7}g_{22}0-g_{-31}-g_{-27}000g_{-18}00g_{-13}00g_{-8}0g_{-7}0g_{-2}000h_{6}+h_{5}+h_{4}+h_{3}0000g_{3}-g_{6}000g_{10}0-g_{12}000g_{16}00-g_{17}000g_{22}-g_{22}00-g_{22}g_{22}0-g_{26}0000-g_{28}-g_{30}0000-g_{33}0000-g_{36}00000000g_{40}000g_{41}00000000000
(0, 0, 1, 1, 1, 0, 1)e_{3}+e_{7}g_{23}0-g_{-30}-g_{-26}000g_{-17}00g_{-12}0g_{-8}00g_{-6}g_{-2}000h_{7}+h_{5}+h_{4}+h_{3}0000g_{3}0-g_{7}00g_{10}00-g_{13}00g_{16}000-g_{18}000g_{23}-g_{23}00g_{23}-g_{23}0-g_{27}000-g_{28}0-g_{31}000-g_{33}0000-g_{36}00000000g_{40}000g_{41}000000000000
(0, 0, 0, 1, 1, 1, 1)e_{4}+e_{6}g_{24}000-g_{-25}-g_{-21}0-g_{-16}0g_{-14}0g_{-9}00g_{-5}g_{-3}000h_{7}+h_{6}+h_{5}+h_{4}0000g_{4}g_{6}g_{7}000000000-g_{17}-g_{18}0-g_{19}00000g_{24}-g_{24}g_{24}-g_{24}-g_{24}00-g_{28}0-g_{29}000-g_{32}0000-g_{34}0g_{36}0000g_{38}000g_{39}00000000000000000
(1, 1, 1, 1, 1, 0, 0)e_{1}-e_{6}g_{25}g_{-32}g_{-28}0g_{-24}0g_{-19}00000-g_{-7}-g_{-6}0000h_{5}+h_{4}+h_{3}+h_{2}+h_{1}000g_{1}-g_{5}000g_{8}0-g_{11}00g_{14}00-g_{16}00g_{20}000-g_{21}-g_{25}000-g_{25}g_{25}g_{25}00000g_{30}g_{31}00000000000-g_{37}0000-g_{39}000-g_{41}000-g_{42}0000000000
(0, 1, 1, 1, 1, 1, 0)e_{2}-e_{7}g_{26}-g_{-31}0g_{-23}0g_{-18}00g_{-13}00g_{-7}0g_{-1}000h_{6}+h_{5}+h_{4}+h_{3}+h_{2}000g_{2}-g_{6}000g_{9}0-g_{12}00g_{15}00-g_{17}00g_{21}000-g_{22}0g_{26}-g_{26}000-g_{26}g_{26}-g_{30}00000-g_{32}00000-g_{35}0000-g_{38}0000-g_{40}0000000g_{42}00000000000
(0, 1, 1, 1, 1, 0, 1)e_{2}+e_{7}g_{27}-g_{-30}0g_{-22}0g_{-17}00g_{-12}00g_{-6}g_{-1}000h_{7}+h_{5}+h_{4}+h_{3}+h_{2}000g_{2}0-g_{7}00g_{9}00-g_{13}0g_{15}000-g_{18}0g_{21}0000-g_{23}0g_{27}-g_{27}000g_{27}-g_{27}-g_{31}0000-g_{32}00000-g_{35}0000-g_{38}0000-g_{40}0000000g_{42}000000000000
(0, 0, 1, 1, 1, 1, 1)e_{3}+e_{6}g_{28}0-g_{-25}-g_{-21}000g_{-11}0g_{-8}g_{-5}g_{-2}000h_{7}+h_{6}+h_{5}+h_{4}+h_{3}000g_{3}g_{6}g_{7}00g_{10}00000000-g_{19}00-g_{22}-g_{23}00-g_{24}000g_{28}-g_{28}0g_{28}-g_{28}-g_{28}0-g_{32}00-g_{33}00-g_{34}00-g_{36}000000000g_{40}000g_{41}00000000000000000
(0, 0, 0, 1, 2, 1, 1)e_{4}+e_{5}g_{29}000-g_{-20}-g_{-15}g_{-14}-g_{-10}g_{-9}0g_{-3}000h_{7}+h_{6}+2h_{5}+h_{4}0000g_{5}0000-g_{11}g_{12}g_{13}000-g_{17}-g_{18}g_{19}00000-g_{24}000000g_{29}0-g_{29}0000-g_{33}00000-g_{35}g_{36}000-g_{37}g_{38}0000g_{39}0000000000000000000000
(1, 1, 1, 1, 1, 1, 0)e_{1}-e_{7}g_{30}g_{-27}g_{-23}0g_{-18}0g_{-13}00g_{-7}000h_{6}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}000g_{1}-g_{6}00g_{8}0-g_{12}00g_{14}00-g_{17}0g_{20}000-g_{22}0g_{25}0000-g_{26}-g_{30}0000-g_{30}g_{30}000000-g_{34}00000-g_{37}0000-g_{39}0000-g_{41}000-g_{42}000000000000000
(1, 1, 1, 1, 1, 0, 1)e_{1}+e_{7}g_{31}g_{-26}g_{-22}0g_{-17}0g_{-12}00g_{-6}00h_{7}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}000g_{1}0-g_{7}0g_{8}00-g_{13}0g_{14}000-g_{18}g_{20}0000-g_{23}g_{25}00000-g_{27}-g_{31}0000g_{31}-g_{31}00000-g_{34}00000-g_{37}0000-g_{39}0000-g_{41}000-g_{42}0000000000000000
(0, 1, 1, 1, 1, 1, 1)e_{2}+e_{6}g_{32}-g_{-25}0g_{-16}0g_{-11}00g_{-5}g_{-1}0h_{7}+h_{6}+h_{5}+h_{4}+h_{3}+h_{2}000g_{2}g_{6}g_{7}0g_{9}0000g_{15}000-g_{19}00000-g_{24}0-g_{26}-g_{27}000-g_{28}0g_{32}-g_{32}00g_{32}-g_{32}-g_{32}-g_{34}000-g_{35}00000-g_{38}0000-g_{40}00000000g_{42}00000000000000000
(0, 0, 1, 1, 2, 1, 1)e_{3}+e_{5}g_{33}0-g_{-20}-g_{-15}00g_{-8}g_{-4}g_{-2}0h_{7}+h_{6}+2h_{5}+h_{4}+h_{3}000g_{3}g_{5}0000g_{12}g_{13}00-g_{16}00g_{19}00-g_{22}-g_{23}000000-g_{28}0-g_{29}000g_{33}-g_{33}g_{33}-g_{33}000-g_{35}0-g_{36}000-g_{37}000000g_{40}0000g_{41}0000000000000000000000
(1, 1, 1, 1, 1, 1, 1)e_{1}+e_{6}g_{34}g_{-21}g_{-16}0g_{-11}0g_{-5}00h_{7}+h_{6}+h_{5}+h_{4}+h_{3}+h_{2}+h_{1}0g_{1}g_{6}g_{7}0g_{8}000g_{14}000-g_{19}g_{20}0000-g_{24}00000-g_{28}-g_{30}-g_{31}0000-g_{32}-g_{34}000g_{34}-g_{34}-g_{34}0000-g_{37}00000-g_{39}0000-g_{41}0000-g_{42}000000000000000000000
(0, 1, 1, 1, 2, 1, 1)e_{2}+e_{5}g_{35}-g_{-20}0g_{-10}0g_{-4}g_{-1}0h_{7}+h_{6}+2h_{5}+h_{4}+h_{3}+h_{2}0g_{2}g_{5}00g_{9}0g_{12}g_{13}0000g_{19}0-g_{21}00000-g_{26}-g_{27}00-g_{29}000-g_{32}00-g_{33}0g_{35}-g_{35}0g_{35}-g_{35}00-g_{37}00-g_{38}00000-g_{40}000000000g_{42}0000000000000000000000
(0, 0, 1, 2, 2, 1, 1)e_{3}+e_{4}g_{36}0-g_{-14}-g_{-9}g_{-8}g_{-2}0h_{7}+h_{6}+2h_{5}+2h_{4}+h_{3}00g_{4}000-g_{10}g_{11}000-g_{16}g_{17}g_{18}000-g_{22}-g_{23}g_{24}0000-g_{28}g_{29}00000-g_{33}0000g_{36}0-g_{36}0000-g_{38}00000-g_{39}g_{40}0000g_{41}0000000000000000000000000000
(1, 1, 1, 1, 2, 1, 1)e_{1}+e_{5}g_{37}g_{-15}g_{-10}0g_{-4}0h_{7}+h_{6}+2h_{5}+h_{4}+h_{3}+h_{2}+h_{1}0g_{1}g_{5}g_{8}0g_{12}g_{13}g_{14}000g_{19}00000-g_{25}0000-g_{29}-g_{30}-g_{31}000-g_{33}00-g_{34}000-g_{35}-g_{37}00g_{37}-g_{37}00000-g_{39}00000-g_{41}0000-g_{42}000000000000000000000000000
(0, 1, 1, 2, 2, 1, 1)e_{2}+e_{4}g_{38}-g_{-14}0g_{-3}g_{-1}h_{7}+h_{6}+2h_{5}+2h_{4}+h_{3}+h_{2}0g_{2}g_{4}00g_{11}00-g_{15}0g_{17}g_{18}0-g_{21}00g_{24}00-g_{26}-g_{27}0g_{29}000-g_{32}000000-g_{35}0-g_{36}0g_{38}-g_{38}g_{38}-g_{38}000-g_{39}0-g_{40}0000000000g_{42}0000000000000000000000000000
(1, 1, 1, 2, 2, 1, 1)e_{1}+e_{4}g_{39}g_{-9}g_{-3}0h_{7}+h_{6}+2h_{5}+2h_{4}+h_{3}+h_{2}+h_{1}g_{1}g_{4}g_{8}0g_{11}00g_{17}g_{18}-g_{20}000g_{24}-g_{25}000g_{29}0-g_{30}-g_{31}00000-g_{34}00-g_{36}000-g_{37}00-g_{38}-g_{39}0g_{39}-g_{39}00000-g_{41}00000-g_{42}000000000000000000000000000000000
(0, 1, 2, 2, 2, 1, 1)e_{2}+e_{3}g_{40}-g_{-8}g_{-1}h_{7}+h_{6}+2h_{5}+2h_{4}+2h_{3}+h_{2}0g_{3}0-g_{9}g_{10}0-g_{15}g_{16}000-g_{21}g_{22}g_{23}00-g_{26}-g_{27}g_{28}0000-g_{32}g_{33}0000-g_{35}g_{36}00000-g_{38}00g_{40}0-g_{40}0000-g_{41}000000g_{42}0000000000000000000000000000000000
(1, 1, 2, 2, 2, 1, 1)e_{1}+e_{3}g_{41}g_{-2}h_{7}+h_{6}+2h_{5}+2h_{4}+2h_{3}+h_{2}+h_{1}g_{1}g_{3}0g_{10}-g_{14}0g_{16}-g_{20}0g_{22}g_{23}0-g_{25}00g_{28}0-g_{30}-g_{31}0g_{33}000-g_{34}0g_{36}000-g_{37}000000-g_{39}0-g_{40}-g_{41}g_{41}-g_{41}00000-g_{42}0000000000000000000000000000000000000000
(1, 2, 2, 2, 2, 1, 1)e_{1}+e_{2}g_{42}h_{7}+h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}g_{2}-g_{8}g_{9}-g_{14}g_{15}0-g_{20}g_{21}0-g_{25}g_{26}g_{27}00-g_{30}-g_{31}g_{32}000-g_{34}g_{35}0000-g_{37}g_{38}0000-g_{39}g_{40}00000-g_{41}00-g_{42}00000000000000000000000000000000000000000000000
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:
\(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0 & 0 & 0\\ -1 & 2 & -1 & 0 & 0 & 0 & 0\\ 0 & -1 & 2 & -1 & 0 & 0 & 0\\ 0 & 0 & -1 & 2 & -1 & 0 & 0\\ 0 & 0 & 0 & -1 & 2 & -1 & -1\\ 0 & 0 & 0 & 0 & -1 & 2 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 2\\ \end{pmatrix}\)
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:
\(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0 & 0 & 0\\ -1 & 2 & -1 & 0 & 0 & 0 & 0\\ 0 & -1 & 2 & -1 & 0 & 0 & 0\\ 0 & 0 & -1 & 2 & -1 & 0 & 0\\ 0 & 0 & 0 & -1 & 2 & -1 & -1\\ 0 & 0 & 0 & 0 & -1 & 2 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 2\\ \end{pmatrix}\)
The determinant of the symmetric Cartan matrix is: 4
Half sum of positive roots: (6, 11, 15, 18, 20, 21/2, 21/2)= \(\displaystyle 6\varepsilon_{1}+5\varepsilon_{2}+4\varepsilon_{3}+3\varepsilon_{4}+2\varepsilon_{5}+\varepsilon_{6}\)
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, 1/2, 1/2) = \(\displaystyle \varepsilon_{1}\)
(1, 2, 2, 2, 2, 1, 1) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}\)
(1, 2, 3, 3, 3, 3/2, 3/2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}\)
(1, 2, 3, 4, 4, 2, 2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}\)
(1, 2, 3, 4, 5, 5/2, 5/2) = \(\displaystyle \varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}\)
(1/2, 1, 3/2, 2, 5/2, 7/4, 5/4) = \(\displaystyle 1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}-1/2\varepsilon_{7}\)
(1/2, 1, 3/2, 2, 5/2, 5/4, 7/4) = \(\displaystyle 1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}+1/2\varepsilon_{4}+1/2\varepsilon_{5}+1/2\varepsilon_{6}+1/2\varepsilon_{7}\)
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) = \(\displaystyle \varepsilon_{1}-\varepsilon_{2}\)
(0, 1, 0, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{2}-\varepsilon_{3}\)
(0, 0, 1, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{3}-\varepsilon_{4}\)
(0, 0, 0, 1, 0, 0, 0) = \(\displaystyle \varepsilon_{4}-\varepsilon_{5}\)
(0, 0, 0, 0, 1, 0, 0) = \(\displaystyle \varepsilon_{5}-\varepsilon_{6}\)
(0, 0, 0, 0, 0, 1, 0) = \(\displaystyle \varepsilon_{6}-\varepsilon_{7}\)
(0, 0, 0, 0, 0, 0, 1) = \(\displaystyle \varepsilon_{6}+\varepsilon_{7}\)