Type: \(\displaystyle 4A^{1}_1\) (Dynkin type computed to be: \(\displaystyle 4A^{1}_1\))
Simple basis: 4 vectors: (1, 2, 2, 2, 2, 2, 1, 1), (0, 0, 1, 2, 2, 2, 1, 1), (0, 0, 0, 0, 1, 2, 1, 1), (0, 0, 0, 0, 0, 0, 1, 0)
Simple basis epsilon form:
Simple basis epsilon form with respect to k:
Number of outer autos with trivial action on orthogonal complement and extending to autos of ambient algebra: 0
Number of outer autos with trivial action on orthogonal complement: 0.
C(k_{ss})_{ss}: 4A^{1}_1
simple basis centralizer: 4 vectors: (0, 0, 0, 0, 0, 0, 0, 1), (0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0, 0), (1, 0, 0, 0, 0, 0, 0, 0)
Number of k-submodules of g: 40
Module decomposition, fundamental coords over k: \(\displaystyle V_{2\omega_{4}}+4V_{\omega_{3}+\omega_{4}}+4V_{\omega_{2}+\omega_{4}}+4V_{\omega_{1}+\omega_{4}}+V_{2\omega_{3}}+4V_{\omega_{2}+\omega_{3}}+4V_{\omega_{1}+\omega_{3}}+V_{2\omega_{2}}+4V_{\omega_{1}+\omega_{2}}+V_{2\omega_{1}}+12V_{0}\)
g/k k-submodules
idsizeb\cap k-lowest weightb\cap k-highest weightModule basisWeights epsilon coords
Module 11(0, 0, 0, 0, 0, 0, 0, -1)(0, 0, 0, 0, 0, 0, 0, -1)g_{-8}-\varepsilon_{7}-\varepsilon_{8}
Module 21(0, 0, 0, 0, -1, 0, 0, 0)(0, 0, 0, 0, -1, 0, 0, 0)g_{-5}-\varepsilon_{5}+\varepsilon_{6}
Module 31(0, 0, -1, 0, 0, 0, 0, 0)(0, 0, -1, 0, 0, 0, 0, 0)g_{-3}-\varepsilon_{3}+\varepsilon_{4}
Module 41(-1, 0, 0, 0, 0, 0, 0, 0)(-1, 0, 0, 0, 0, 0, 0, 0)g_{-1}-\varepsilon_{1}+\varepsilon_{2}
Module 51(1, 0, 0, 0, 0, 0, 0, 0)(1, 0, 0, 0, 0, 0, 0, 0)g_{1}\varepsilon_{1}-\varepsilon_{2}
Module 61(0, 0, 1, 0, 0, 0, 0, 0)(0, 0, 1, 0, 0, 0, 0, 0)g_{3}\varepsilon_{3}-\varepsilon_{4}
Module 71(0, 0, 0, 0, 1, 0, 0, 0)(0, 0, 0, 0, 1, 0, 0, 0)g_{5}\varepsilon_{5}-\varepsilon_{6}
Module 83(0, 0, 0, 0, 0, 0, -1, 0)(0, 0, 0, 0, 0, 0, 1, 0)g_{7}
h_{7}
g_{-7}		\varepsilon_{7}-\varepsilon_{8}
0
-\varepsilon_{7}+\varepsilon_{8}
Module 91(0, 0, 0, 0, 0, 0, 0, 1)(0, 0, 0, 0, 0, 0, 0, 1)g_{8}\varepsilon_{7}+\varepsilon_{8}
Module 104(0, 0, 0, 0, -1, -1, -1, -1)(0, 0, 0, 0, 0, 1, 1, 0)g_{14}
g_{-21}
g_{6}
g_{-28}		\varepsilon_{6}-\varepsilon_{8}
-\varepsilon_{5}-\varepsilon_{8}
\varepsilon_{6}-\varepsilon_{7}
-\varepsilon_{5}-\varepsilon_{7}
Module 114(0, 0, 0, 0, 0, -1, -1, -1)(0, 0, 0, 0, 1, 1, 1, 0)g_{20}
g_{-15}
g_{13}
g_{-22}		\varepsilon_{5}-\varepsilon_{8}
-\varepsilon_{6}-\varepsilon_{8}
\varepsilon_{5}-\varepsilon_{7}
-\varepsilon_{6}-\varepsilon_{7}
Module 124(0, 0, 0, 0, -1, -1, -1, 0)(0, 0, 0, 0, 0, 1, 1, 1)g_{22}
g_{-13}
g_{15}
g_{-20}		\varepsilon_{6}+\varepsilon_{7}
-\varepsilon_{5}+\varepsilon_{7}
\varepsilon_{6}+\varepsilon_{8}
-\varepsilon_{5}+\varepsilon_{8}
Module 134(0, 0, -1, -1, -1, -1, -1, -1)(0, 0, 0, 1, 1, 1, 1, 0)g_{26}
g_{-32}
g_{19}
g_{-38}		\varepsilon_{4}-\varepsilon_{8}
-\varepsilon_{3}-\varepsilon_{8}
\varepsilon_{4}-\varepsilon_{7}
-\varepsilon_{3}-\varepsilon_{7}
Module 144(0, 0, 0, 0, 0, -1, -1, 0)(0, 0, 0, 0, 1, 1, 1, 1)g_{28}
g_{-6}
g_{21}
g_{-14}		\varepsilon_{5}+\varepsilon_{7}
-\varepsilon_{6}+\varepsilon_{7}
\varepsilon_{5}+\varepsilon_{8}
-\varepsilon_{6}+\varepsilon_{8}
Module 154(0, 0, 0, -1, -1, -1, -1, -1)(0, 0, 1, 1, 1, 1, 1, 0)g_{31}
g_{-27}
g_{25}
g_{-33}		\varepsilon_{3}-\varepsilon_{8}
-\varepsilon_{4}-\varepsilon_{8}
\varepsilon_{3}-\varepsilon_{7}
-\varepsilon_{4}-\varepsilon_{7}
Module 164(0, 0, -1, -1, -1, -1, -1, 0)(0, 0, 0, 1, 1, 1, 1, 1)g_{33}
g_{-25}
g_{27}
g_{-31}		\varepsilon_{4}+\varepsilon_{7}
-\varepsilon_{3}+\varepsilon_{7}
\varepsilon_{4}+\varepsilon_{8}
-\varepsilon_{3}+\varepsilon_{8}
Module 173(0, 0, 0, 0, -1, -2, -1, -1)(0, 0, 0, 0, 1, 2, 1, 1)g_{34}
h_{8}+h_{7}+2h_{6}+h_{5}
g_{-34}		\varepsilon_{5}+\varepsilon_{6}
0
-\varepsilon_{5}-\varepsilon_{6}
Module 184(-1, -1, -1, -1, -1, -1, -1, -1)(0, 1, 1, 1, 1, 1, 1, 0)g_{36}
g_{-41}
g_{30}
g_{-45}		\varepsilon_{2}-\varepsilon_{8}
-\varepsilon_{1}-\varepsilon_{8}
\varepsilon_{2}-\varepsilon_{7}
-\varepsilon_{1}-\varepsilon_{7}
Module 194(0, 0, 0, -1, -1, -1, -1, 0)(0, 0, 1, 1, 1, 1, 1, 1)g_{38}
g_{-19}
g_{32}
g_{-26}		\varepsilon_{3}+\varepsilon_{7}
-\varepsilon_{4}+\varepsilon_{7}
\varepsilon_{3}+\varepsilon_{8}
-\varepsilon_{4}+\varepsilon_{8}
Module 204(0, 0, -1, -1, -2, -2, -1, -1)(0, 0, 0, 1, 1, 2, 1, 1)g_{39}
g_{-18}
g_{4}
g_{-47}		\varepsilon_{4}+\varepsilon_{6}
-\varepsilon_{3}+\varepsilon_{6}
\varepsilon_{4}-\varepsilon_{5}
-\varepsilon_{3}-\varepsilon_{5}
Module 214(0, -1, -1, -1, -1, -1, -1, -1)(1, 1, 1, 1, 1, 1, 1, 0)g_{40}
g_{-37}
g_{35}
g_{-42}		\varepsilon_{1}-\varepsilon_{8}
-\varepsilon_{2}-\varepsilon_{8}
\varepsilon_{1}-\varepsilon_{7}
-\varepsilon_{2}-\varepsilon_{7}
Module 224(-1, -1, -1, -1, -1, -1, -1, 0)(0, 1, 1, 1, 1, 1, 1, 1)g_{42}
g_{-35}
g_{37}
g_{-40}		\varepsilon_{2}+\varepsilon_{7}
-\varepsilon_{1}+\varepsilon_{7}
\varepsilon_{2}+\varepsilon_{8}
-\varepsilon_{1}+\varepsilon_{8}
Module 234(0, 0, 0, -1, -2, -2, -1, -1)(0, 0, 1, 1, 1, 2, 1, 1)g_{43}
g_{-12}
g_{11}
g_{-44}		\varepsilon_{3}+\varepsilon_{6}
-\varepsilon_{4}+\varepsilon_{6}
\varepsilon_{3}-\varepsilon_{5}
-\varepsilon_{4}-\varepsilon_{5}
Module 244(0, 0, -1, -1, -1, -2, -1, -1)(0, 0, 0, 1, 2, 2, 1, 1)g_{44}
g_{-11}
g_{12}
g_{-43}		\varepsilon_{4}+\varepsilon_{5}
-\varepsilon_{3}+\varepsilon_{5}
\varepsilon_{4}-\varepsilon_{6}
-\varepsilon_{3}-\varepsilon_{6}
Module 254(0, -1, -1, -1, -1, -1, -1, 0)(1, 1, 1, 1, 1, 1, 1, 1)g_{45}
g_{-30}
g_{41}
g_{-36}		\varepsilon_{1}+\varepsilon_{7}
-\varepsilon_{2}+\varepsilon_{7}
\varepsilon_{1}+\varepsilon_{8}
-\varepsilon_{2}+\varepsilon_{8}
Module 264(-1, -1, -1, -1, -2, -2, -1, -1)(0, 1, 1, 1, 1, 2, 1, 1)g_{46}
g_{-29}
g_{17}
g_{-51}		\varepsilon_{2}+\varepsilon_{6}
-\varepsilon_{1}+\varepsilon_{6}
\varepsilon_{2}-\varepsilon_{5}
-\varepsilon_{1}-\varepsilon_{5}
Module 274(0, 0, 0, -1, -1, -2, -1, -1)(0, 0, 1, 1, 2, 2, 1, 1)g_{47}
g_{-4}
g_{18}
g_{-39}		\varepsilon_{3}+\varepsilon_{5}
-\varepsilon_{4}+\varepsilon_{5}
\varepsilon_{3}-\varepsilon_{6}
-\varepsilon_{4}-\varepsilon_{6}
Module 284(0, -1, -1, -1, -2, -2, -1, -1)(1, 1, 1, 1, 1, 2, 1, 1)g_{48}
g_{-24}
g_{23}
g_{-49}		\varepsilon_{1}+\varepsilon_{6}
-\varepsilon_{2}+\varepsilon_{6}
\varepsilon_{1}-\varepsilon_{5}
-\varepsilon_{2}-\varepsilon_{5}
Module 294(-1, -1, -1, -1, -1, -2, -1, -1)(0, 1, 1, 1, 2, 2, 1, 1)g_{49}
g_{-23}
g_{24}
g_{-48}		\varepsilon_{2}+\varepsilon_{5}
-\varepsilon_{1}+\varepsilon_{5}
\varepsilon_{2}-\varepsilon_{6}
-\varepsilon_{1}-\varepsilon_{6}
Module 303(0, 0, -1, -2, -2, -2, -1, -1)(0, 0, 1, 2, 2, 2, 1, 1)g_{50}
h_{8}+h_{7}+2h_{6}+2h_{5}+2h_{4}+h_{3}
g_{-50}		\varepsilon_{3}+\varepsilon_{4}
0
-\varepsilon_{3}-\varepsilon_{4}
Module 314(0, -1, -1, -1, -1, -2, -1, -1)(1, 1, 1, 1, 2, 2, 1, 1)g_{51}
g_{-17}
g_{29}
g_{-46}		\varepsilon_{1}+\varepsilon_{5}
-\varepsilon_{2}+\varepsilon_{5}
\varepsilon_{1}-\varepsilon_{6}
-\varepsilon_{2}-\varepsilon_{6}
Module 324(-1, -1, -2, -2, -2, -2, -1, -1)(0, 1, 1, 2, 2, 2, 1, 1)g_{52}
g_{-16}
g_{2}
g_{-55}		\varepsilon_{2}+\varepsilon_{4}
-\varepsilon_{1}+\varepsilon_{4}
\varepsilon_{2}-\varepsilon_{3}
-\varepsilon_{1}-\varepsilon_{3}
Module 334(0, -1, -2, -2, -2, -2, -1, -1)(1, 1, 1, 2, 2, 2, 1, 1)g_{53}
g_{-10}
g_{9}
g_{-54}		\varepsilon_{1}+\varepsilon_{4}
-\varepsilon_{2}+\varepsilon_{4}
\varepsilon_{1}-\varepsilon_{3}
-\varepsilon_{2}-\varepsilon_{3}
Module 344(-1, -1, -1, -2, -2, -2, -1, -1)(0, 1, 2, 2, 2, 2, 1, 1)g_{54}
g_{-9}
g_{10}
g_{-53}		\varepsilon_{2}+\varepsilon_{3}
-\varepsilon_{1}+\varepsilon_{3}
\varepsilon_{2}-\varepsilon_{4}
-\varepsilon_{1}-\varepsilon_{4}
Module 354(0, -1, -1, -2, -2, -2, -1, -1)(1, 1, 2, 2, 2, 2, 1, 1)g_{55}
g_{-2}
g_{16}
g_{-52}		\varepsilon_{1}+\varepsilon_{3}
-\varepsilon_{2}+\varepsilon_{3}
\varepsilon_{1}-\varepsilon_{4}
-\varepsilon_{2}-\varepsilon_{4}
Module 363(-1, -2, -2, -2, -2, -2, -1, -1)(1, 2, 2, 2, 2, 2, 1, 1)g_{56}
h_{8}+h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+h_{1}
g_{-56}		\varepsilon_{1}+\varepsilon_{2}
0
-\varepsilon_{1}-\varepsilon_{2}
Module 371(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{1}0
Module 381(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{3}0
Module 391(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{5}0
Module 401(0, 0, 0, 0, 0, 0, 0, 0)(0, 0, 0, 0, 0, 0, 0, 0)h_{8}0
Information about the subalgebra generation algorithm.
Heirs rejected due to having symmetric Cartan type outside of list dictated by parabolic heirs: 28
Heirs rejected due to not being maximally dominant: 4
Heirs rejected due to not being maximal with respect to small Dynkin diagram automorphism that extends to ambient automorphism: 4
Heirs rejected due to having ambient Lie algebra decomposition iso to an already found subalgebra: 4
Parabolically induced by 3A^{1}_1
Potential Dynkin type extensions: 5A^{1}_1,