Subalgebra

$A^{2}_2+A^{28}_1$ ↪

$E^{1}_6$
82 out of 119

Computations done by the calculator project.

Subalgebra type:

$\displaystyle A^{2}_2+A^{28}_1$ (click on type for detailed printout).
Subalgebra is (parabolically) induced from

$\displaystyle A^{2}_2$ .
Centralizer: 0
The semisimple part of the centralizer of the semisimple part of my centralizer:

$\displaystyle E^{1}_6$

Elements Cartan subalgebra scaled to act by two by components:

$\displaystyle A^{2}_2$ : (2, 2, 3, 4, 3, 2): 4, (0, -1, -1, -2, -2, -2): 4,

$\displaystyle A^{28}_1$ : (0, 6, 6, 10, 6, 0): 56
Dimension of subalgebra generated by predefined or computed generators: 11.
Negative simple generators:

$\displaystyle g_{-32}+g_{-33}$ ,

$\displaystyle g_{21}+g_{20}$ ,

$\displaystyle g_{-2}-g_{-3}+g_{-4}+g_{-5}$
Positive simple generators:

$\displaystyle g_{33}+g_{32}$ ,

$\displaystyle g_{-20}+g_{-21}$ ,

$\displaystyle 6g_{5}+10g_{4}-6g_{3}+6g_{2}$
Cartan symmetric matrix:

$\displaystyle \begin{pmatrix}1 & -1/2 & 0\\ -1/2 & 1 & 0\\ 0 & 0 & 1/14\\ \end{pmatrix}$
Scalar products of elements of Cartan subalgebra scaled to act by 2 (co-symmetric Cartan matrix):

$\displaystyle \begin{pmatrix}4 & -2 & 0\\ -2 & 4 & 0\\ 0 & 0 & 56\\ \end{pmatrix}$
Decomposition of ambient Lie algebra:

$\displaystyle V_{10\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}+6\omega_{3}}\oplus V_{2\omega_{3}}\oplus V_{\omega_{1}+\omega_{2}}$
In the table below we indicate the highest weight vectors of the decomposition of the ambient Lie algebra as a module over the semisimple part. The second row indicates weights of the highest weight vectors relative to the Cartan of the semisimple subalgebra.

Highest vectors of representations (total 4) ; the vectors are over the primal subalgebra.	$g_{18}+g_{17}$	$g_{5}+5/3g_{4}-g_{3}+g_{2}$	$g_{29}$	$g_{23}$
weight	$\omega_{1}+\omega_{2}$	$2\omega_{3}$	$\omega_{1}+\omega_{2}+6\omega_{3}$	$10\omega_{3}$

Isotypic module decomposition over primal subalgebra (total 4 isotypic components).

Isotypical components + highest weight

$\displaystyle V_{\omega_{1}+\omega_{2}}$ → (1, 1, 0)

$\displaystyle V_{2\omega_{3}}$ → (0, 0, 2)

$\displaystyle V_{\omega_{1}+\omega_{2}+6\omega_{3}}$ → (1, 1, 6)

$\displaystyle V_{10\omega_{3}}$ → (0, 0, 10)

Module label

$W_{1}$

$W_{2}$

$W_{3}$

$W_{4}$

Module elements (weight vectors). In blue - corresp. F element. In red -corresp. H element.

Semisimple subalgebra component.

$-g_{18}-g_{17}$

$-g_{-20}-g_{-21}$

$g_{33}+g_{32}$

$-2h_{6}-2h_{5}-2h_{4}-h_{3}-h_{2}$

$-2h_{6}-3h_{5}-4h_{4}-3h_{3}-2h_{2}-2h_{1}$

$-g_{-32}-g_{-33}$

$2g_{21}+2g_{20}$

$-g_{-17}-g_{-18}$

Semisimple subalgebra component.

$-g_{5}-5/3g_{4}+g_{3}-g_{2}$

$h_{5}+5/3h_{4}+h_{3}+h_{2}$

$1/3g_{-2}-1/3g_{-3}+1/3g_{-4}+1/3g_{-5}$

$g_{29}$

$g_{-6}$

$-g_{36}$

$-g_{26}$

$g_{15}+g_{14}$

$-g_{-11}$

$g_{14}-g_{13}$

$-g_{35}$

$-g_{22}$

$g_{-24}$

$-2g_{31}$

$-g_{9}-g_{8}$

$g_{-16}$

$g_{10}-g_{9}$

$-g_{34}$

$-g_{18}+g_{17}$

$g_{-1}$

$-g_{-27}$

$-2g_{28}$

$g_{3}+g_{2}$

$-g_{-20}+g_{-21}$

$g_{5}+g_{3}$

$g_{33}-g_{32}$

$2g_{12}$

$-g_{-7}$

$g_{-30}$

$-2g_{25}$

$h_{3}-h_{2}$

$-2g_{-25}$

$-h_{5}+h_{3}$

$2g_{30}$

$-2g_{7}$

$-g_{-12}$

$g_{-32}-g_{-33}$

$-2g_{21}+2g_{20}$

$-2g_{-2}-2g_{-3}$

$2g_{-28}$

$-2g_{-3}-2g_{-5}$

$2g_{27}$

$-2g_{1}$

$g_{-17}-g_{-18}$

$-2g_{-34}$

$4g_{16}$

$-2g_{-8}-2g_{-9}$

$-2g_{-31}$

$-2g_{-9}+2g_{-10}$

$2g_{24}$

$2g_{-22}$

$2g_{-35}$

$4g_{11}$

$-2g_{-14}-2g_{-15}$

$2g_{-13}-2g_{-14}$

$-2g_{-26}$

$-2g_{-36}$

$4g_{6}$

$-2g_{-29}$

$g_{23}$

$g_{19}$

$g_{15}-g_{14}-g_{13}$

$-2g_{10}-2g_{9}+2g_{8}$

$-2g_{5}+6g_{4}+2g_{3}-2g_{2}$

$2h_{5}-6h_{4}+2h_{3}+2h_{2}$

$10g_{-2}-10g_{-3}-18g_{-4}+10g_{-5}$

$28g_{-8}-28g_{-9}-28g_{-10}$

$56g_{-13}+56g_{-14}-56g_{-15}$

$168g_{-19}$

$-168g_{-23}$

Weights of elements in fundamental coords w.r.t. Cartan of subalgebra in same order as above

$\omega_{1}+\omega_{2}$

$-\omega_{1}+2\omega_{2}$

$2\omega_{1}-\omega_{2}$

$0$

$-2\omega_{1}+\omega_{2}$

$\omega_{1}-2\omega_{2}$

$-\omega_{1}-\omega_{2}$

$2\omega_{3}$

$0$

$-2\omega_{3}$

$\omega_{1}+\omega_{2}+6\omega_{3}$

$-\omega_{1}+2\omega_{2}+6\omega_{3}$

$2\omega_{1}-\omega_{2}+6\omega_{3}$

$\omega_{1}+\omega_{2}+4\omega_{3}$

$6\omega_{3}$

$-\omega_{1}+2\omega_{2}+4\omega_{3}$

$6\omega_{3}$

$2\omega_{1}-\omega_{2}+4\omega_{3}$

$\omega_{1}+\omega_{2}+2\omega_{3}$

$-2\omega_{1}+\omega_{2}+6\omega_{3}$

$\omega_{1}-2\omega_{2}+6\omega_{3}$

$4\omega_{3}$

$-\omega_{1}+2\omega_{2}+2\omega_{3}$

$4\omega_{3}$

$2\omega_{1}-\omega_{2}+2\omega_{3}$

$\omega_{1}+\omega_{2}$

$-\omega_{1}-\omega_{2}+6\omega_{3}$

$-2\omega_{1}+\omega_{2}+4\omega_{3}$

$\omega_{1}-2\omega_{2}+4\omega_{3}$

$2\omega_{3}$

$-\omega_{1}+2\omega_{2}$

$2\omega_{3}$

$2\omega_{1}-\omega_{2}$

$\omega_{1}+\omega_{2}-2\omega_{3}$

$-\omega_{1}-\omega_{2}+4\omega_{3}$

$-2\omega_{1}+\omega_{2}+2\omega_{3}$

$\omega_{1}-2\omega_{2}+2\omega_{3}$

$0$

$-\omega_{1}+2\omega_{2}-2\omega_{3}$

$0$

$2\omega_{1}-\omega_{2}-2\omega_{3}$

$\omega_{1}+\omega_{2}-4\omega_{3}$

$-\omega_{1}-\omega_{2}+2\omega_{3}$

$-2\omega_{1}+\omega_{2}$

$\omega_{1}-2\omega_{2}$

$-2\omega_{3}$

$-\omega_{1}+2\omega_{2}-4\omega_{3}$

$-2\omega_{3}$

$2\omega_{1}-\omega_{2}-4\omega_{3}$

$\omega_{1}+\omega_{2}-6\omega_{3}$

$-\omega_{1}-\omega_{2}$

$-2\omega_{1}+\omega_{2}-2\omega_{3}$

$\omega_{1}-2\omega_{2}-2\omega_{3}$

$-4\omega_{3}$

$-\omega_{1}+2\omega_{2}-6\omega_{3}$

$-4\omega_{3}$

$2\omega_{1}-\omega_{2}-6\omega_{3}$

$-\omega_{1}-\omega_{2}-2\omega_{3}$

$-2\omega_{1}+\omega_{2}-4\omega_{3}$

$\omega_{1}-2\omega_{2}-4\omega_{3}$

$-6\omega_{3}$

$-\omega_{1}-\omega_{2}-4\omega_{3}$

$-2\omega_{1}+\omega_{2}-6\omega_{3}$

$\omega_{1}-2\omega_{2}-6\omega_{3}$

$-\omega_{1}-\omega_{2}-6\omega_{3}$

$10\omega_{3}$

$8\omega_{3}$

$6\omega_{3}$

$4\omega_{3}$

$2\omega_{3}$

$0$

$-2\omega_{3}$

$-4\omega_{3}$

$-6\omega_{3}$

$-8\omega_{3}$

$-10\omega_{3}$

Weights of elements in (fundamental coords w.r.t. Cartan of subalgebra) + Cartan centralizer

$\omega_{1}+\omega_{2}$

$-\omega_{1}+2\omega_{2}$

$2\omega_{1}-\omega_{2}$

$0$

$-2\omega_{1}+\omega_{2}$

$\omega_{1}-2\omega_{2}$

$-\omega_{1}-\omega_{2}$

$2\omega_{3}$

$0$

$-2\omega_{3}$

$\omega_{1}+\omega_{2}+6\omega_{3}$

$-\omega_{1}+2\omega_{2}+6\omega_{3}$

$2\omega_{1}-\omega_{2}+6\omega_{3}$

$\omega_{1}+\omega_{2}+4\omega_{3}$

$6\omega_{3}$

$-\omega_{1}+2\omega_{2}+4\omega_{3}$

$6\omega_{3}$

$2\omega_{1}-\omega_{2}+4\omega_{3}$

$\omega_{1}+\omega_{2}+2\omega_{3}$

$-2\omega_{1}+\omega_{2}+6\omega_{3}$

$\omega_{1}-2\omega_{2}+6\omega_{3}$

$4\omega_{3}$

$-\omega_{1}+2\omega_{2}+2\omega_{3}$

$4\omega_{3}$

$2\omega_{1}-\omega_{2}+2\omega_{3}$

$\omega_{1}+\omega_{2}$

$-\omega_{1}-\omega_{2}+6\omega_{3}$

$-2\omega_{1}+\omega_{2}+4\omega_{3}$

$\omega_{1}-2\omega_{2}+4\omega_{3}$

$2\omega_{3}$

$-\omega_{1}+2\omega_{2}$

$2\omega_{3}$

$2\omega_{1}-\omega_{2}$

$\omega_{1}+\omega_{2}-2\omega_{3}$

$-\omega_{1}-\omega_{2}+4\omega_{3}$

$-2\omega_{1}+\omega_{2}+2\omega_{3}$

$\omega_{1}-2\omega_{2}+2\omega_{3}$

$0$

$-\omega_{1}+2\omega_{2}-2\omega_{3}$

$0$

$2\omega_{1}-\omega_{2}-2\omega_{3}$

$\omega_{1}+\omega_{2}-4\omega_{3}$

$-\omega_{1}-\omega_{2}+2\omega_{3}$

$-2\omega_{1}+\omega_{2}$

$\omega_{1}-2\omega_{2}$

$-2\omega_{3}$

$-\omega_{1}+2\omega_{2}-4\omega_{3}$

$-2\omega_{3}$

$2\omega_{1}-\omega_{2}-4\omega_{3}$

$\omega_{1}+\omega_{2}-6\omega_{3}$

$-\omega_{1}-\omega_{2}$

$-2\omega_{1}+\omega_{2}-2\omega_{3}$

$\omega_{1}-2\omega_{2}-2\omega_{3}$

$-4\omega_{3}$

$-\omega_{1}+2\omega_{2}-6\omega_{3}$

$-4\omega_{3}$

$2\omega_{1}-\omega_{2}-6\omega_{3}$

$-\omega_{1}-\omega_{2}-2\omega_{3}$

$-2\omega_{1}+\omega_{2}-4\omega_{3}$

$\omega_{1}-2\omega_{2}-4\omega_{3}$

$-6\omega_{3}$

$-\omega_{1}-\omega_{2}-4\omega_{3}$

$-2\omega_{1}+\omega_{2}-6\omega_{3}$

$\omega_{1}-2\omega_{2}-6\omega_{3}$

$-\omega_{1}-\omega_{2}-6\omega_{3}$

$10\omega_{3}$

$8\omega_{3}$

$6\omega_{3}$

$4\omega_{3}$

$2\omega_{3}$

$0$

$-2\omega_{3}$

$-4\omega_{3}$

$-6\omega_{3}$

$-8\omega_{3}$

$-10\omega_{3}$

Single module character over Cartan of s.a.+ Cartan of centralizer of s.a.

$\displaystyle M_{\omega_{1}+\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus 2M_{0}\oplus M_{-2\omega_{1}+\omega_{2}} \oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}}$

$\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}$

$\displaystyle M_{\omega_{1}+\omega_{2}+6\omega_{3}}\oplus M_{-\omega_{1}+2\omega_{2}+6\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}+6\omega_{3}} \oplus 2M_{6\omega_{3}}\oplus M_{\omega_{1}+\omega_{2}+4\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}+6\omega_{3}}\oplus M_{\omega_{1}-2\omega_{2}+6\omega_{3}} \oplus M_{-\omega_{1}+2\omega_{2}+4\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}+4\omega_{3}}\oplus M_{-\omega_{1}-\omega_{2}+6\omega_{3}} \oplus 2M_{4\omega_{3}}\oplus M_{\omega_{1}+\omega_{2}+2\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}+4\omega_{3}}\oplus M_{\omega_{1}-2\omega_{2}+4\omega_{3}} \oplus M_{-\omega_{1}+2\omega_{2}+2\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}+2\omega_{3}}\oplus M_{-\omega_{1}-\omega_{2}+4\omega_{3}} \oplus 2M_{2\omega_{3}}\oplus M_{\omega_{1}+\omega_{2}}\oplus M_{-2\omega_{1}+\omega_{2}+2\omega_{3}}\oplus M_{\omega_{1}-2\omega_{2}+2\omega_{3}} \oplus M_{-\omega_{1}+2\omega_{2}}\oplus M_{2\omega_{1}-\omega_{2}}\oplus M_{-\omega_{1}-\omega_{2}+2\omega_{3}}\oplus 2M_{0}\oplus M_{\omega_{1}+\omega_{2}-2\omega_{3}} \oplus M_{-2\omega_{1}+\omega_{2}}\oplus M_{\omega_{1}-2\omega_{2}}\oplus M_{-\omega_{1}+2\omega_{2}-2\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}-2\omega_{3}} \oplus M_{-\omega_{1}-\omega_{2}}\oplus 2M_{-2\omega_{3}}\oplus M_{\omega_{1}+\omega_{2}-4\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}-2\omega_{3}} \oplus M_{\omega_{1}-2\omega_{2}-2\omega_{3}}\oplus M_{-\omega_{1}+2\omega_{2}-4\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}-4\omega_{3}} \oplus M_{-\omega_{1}-\omega_{2}-2\omega_{3}}\oplus 2M_{-4\omega_{3}}\oplus M_{\omega_{1}+\omega_{2}-6\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}-4\omega_{3}} \oplus M_{\omega_{1}-2\omega_{2}-4\omega_{3}}\oplus M_{-\omega_{1}+2\omega_{2}-6\omega_{3}}\oplus M_{2\omega_{1}-\omega_{2}-6\omega_{3}} \oplus M_{-\omega_{1}-\omega_{2}-4\omega_{3}}\oplus 2M_{-6\omega_{3}}\oplus M_{-2\omega_{1}+\omega_{2}-6\omega_{3}}\oplus M_{\omega_{1}-2\omega_{2}-6\omega_{3}} \oplus M_{-\omega_{1}-\omega_{2}-6\omega_{3}}$

$\displaystyle M_{10\omega_{3}}\oplus M_{8\omega_{3}}\oplus M_{6\omega_{3}}\oplus M_{4\omega_{3}}\oplus M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}\oplus M_{-4\omega_{3}} \oplus M_{-6\omega_{3}}\oplus M_{-8\omega_{3}}\oplus M_{-10\omega_{3}}$

Isotypic character

$\displaystyle M_{2\omega_{3}}\oplus M_{0}\oplus M_{-2\omega_{3}}$

Semisimple subalgebra: W_{1}+W_{2}
Centralizer extension: 0

Weight diagram. The coordinates corresponding to the simple roots of the subalgerba are fundamental.
The bilinear form is therefore given relative to the fundamental coordinates.

Mouse position: (0.00, 0.00)
Selected index: -1
Coordinate center in screen coordinates:
(200.00, 980.00)
The projection plane (drawn on the screen) is spanned by the following two vectors.
(1.00, 0.00, 0.00)
(0.00, 1.00, 0.00)
0: (1.00, 0.00, 0.00): (333.33, 1046.67)
1: (0.00, 1.00, 0.00): (266.67, 1113.33)
2: (0.00, 0.00, 1.00): (200.00, 980.00)

Made total 6570789 arithmetic operations while solving the Serre relations polynomial system.
The total number of arithmetic operations I needed to solve the Serre relations polynomial system was larger than 1 000 000. I am printing out the Serre relations system for you: maybe that can help improve the polynomial system algorithms.
Subalgebra realized.
3*2 (unknown) gens:
(
x_{1} g_{-32}+x_{2} g_{-33}, x_{10} g_{33}+x_{9} g_{32},
x_{3} g_{21}+x_{4} g_{20}, x_{12} g_{-20}+x_{11} g_{-21},
x_{5} g_{-2}+x_{6} g_{-3}+x_{7} g_{-4}+x_{8} g_{-5}, x_{16} g_{5}+x_{15} g_{4}+x_{14} g_{3}+x_{13} g_{2})
h: (2, 2, 3, 4, 3, 2), e = combination of g_{32} g_{33} , f= combination of g_{-32} g_{-33} h: (0, -1, -1, -2, -2, -2), e = combination of g_{-21} g_{-20} , f= combination of g_{21} g_{20} h: (0, 6, 6, 10, 6, 0), e = combination of g_{2} g_{3} g_{4} g_{5} , f= combination of g_{-2} g_{-3} g_{-4} g_{-5} Positive weight subsystem: 4 vectors: (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0)
Symmetric Cartan default scale: \begin{pmatrix}

2 &	-1 &	0\\
-1 &	2 &	0\\
0 &	0 &	2\\

\end{pmatrix}Character ambient Lie algebra: V_{10\omega_{3}}+V_{8\omega_{3}}+V_{\omega_{1}+\omega_{2}+6\omega_{3}}+V_{-\omega_{1}+2\omega_{2}+6\omega_{3}}+V_{2\omega_{1}-\omega_{2}+6\omega_{3}}+3V_{6\omega_{3}}+V_{\omega_{1}+\omega_{2}+4\omega_{3}}+V_{-2\omega_{1}+\omega_{2}+6\omega_{3}}+V_{\omega_{1}-2\omega_{2}+6\omega_{3}}+V_{-\omega_{1}+2\omega_{2}+4\omega_{3}}+V_{2\omega_{1}-\omega_{2}+4\omega_{3}}+V_{-\omega_{1}-\omega_{2}+6\omega_{3}}+3V_{4\omega_{3}}+V_{\omega_{1}+\omega_{2}+2\omega_{3}}+V_{-2\omega_{1}+\omega_{2}+4\omega_{3}}+V_{\omega_{1}-2\omega_{2}+4\omega_{3}}+V_{-\omega_{1}+2\omega_{2}+2\omega_{3}}+V_{2\omega_{1}-\omega_{2}+2\omega_{3}}+V_{-\omega_{1}-\omega_{2}+4\omega_{3}}+4V_{2\omega_{3}}+2V_{\omega_{1}+\omega_{2}}+V_{-2\omega_{1}+\omega_{2}+2\omega_{3}}+V_{\omega_{1}-2\omega_{2}+2\omega_{3}}+2V_{-\omega_{1}+2\omega_{2}}+2V_{2\omega_{1}-\omega_{2}}+V_{-\omega_{1}-\omega_{2}+2\omega_{3}}+6V_{0}+V_{\omega_{1}+\omega_{2}-2\omega_{3}}+2V_{-2\omega_{1}+\omega_{2}}+2V_{\omega_{1}-2\omega_{2}}+V_{-\omega_{1}+2\omega_{2}-2\omega_{3}}+V_{2\omega_{1}-\omega_{2}-2\omega_{3}}+2V_{-\omega_{1}-\omega_{2}}+4V_{-2\omega_{3}}+V_{\omega_{1}+\omega_{2}-4\omega_{3}}+V_{-2\omega_{1}+\omega_{2}-2\omega_{3}}+V_{\omega_{1}-2\omega_{2}-2\omega_{3}}+V_{-\omega_{1}+2\omega_{2}-4\omega_{3}}+V_{2\omega_{1}-\omega_{2}-4\omega_{3}}+V_{-\omega_{1}-\omega_{2}-2\omega_{3}}+3V_{-4\omega_{3}}+V_{\omega_{1}+\omega_{2}-6\omega_{3}}+V_{-2\omega_{1}+\omega_{2}-4\omega_{3}}+V_{\omega_{1}-2\omega_{2}-4\omega_{3}}+V_{-\omega_{1}+2\omega_{2}-6\omega_{3}}+V_{2\omega_{1}-\omega_{2}-6\omega_{3}}+V_{-\omega_{1}-\omega_{2}-4\omega_{3}}+3V_{-6\omega_{3}}+V_{-2\omega_{1}+\omega_{2}-6\omega_{3}}+V_{\omega_{1}-2\omega_{2}-6\omega_{3}}+V_{-\omega_{1}-\omega_{2}-6\omega_{3}}+V_{-8\omega_{3}}+V_{-10\omega_{3}}
A necessary system to realize the candidate subalgebra.
x_{2} x_{10} +x_{1} x_{9} -2= 0
x_{2} x_{10} +2x_{1} x_{9} -3= 0
2x_{2} x_{10} +x_{1} x_{9} -3= 0
x_{2} x_{16} +x_{1} x_{14} = 0
x_{9} x_{16} +x_{10} x_{14} = 0
x_{1} x_{8} +x_{2} x_{6} = 0
x_{3} x_{11} -1= 0
x_{4} x_{12} +x_{3} x_{11} -2= 0
x_{4} x_{12} -1= 0
x_{4} x_{14} +x_{3} x_{13} = 0
x_{11} x_{14} +x_{12} x_{13} = 0
x_{3} x_{6} +x_{4} x_{5} = 0
x_{5} x_{13} -6= 0
x_{6} x_{14} -6= 0
x_{7} x_{15} -10= 0
x_{8} x_{16} -6= 0
x_{8} x_{10} +x_{6} x_{9} = 0
x_{6} x_{12} +x_{5} x_{11} = 0
The above system after transformation.
x_{2} x_{10} +x_{1} x_{9} -2= 0
x_{2} x_{10} +2x_{1} x_{9} -3= 0
2x_{2} x_{10} +x_{1} x_{9} -3= 0
x_{2} x_{16} +x_{1} x_{14} = 0
x_{9} x_{16} +x_{10} x_{14} = 0
x_{1} x_{8} +x_{2} x_{6} = 0
x_{3} x_{11} -1= 0
x_{4} x_{12} +x_{3} x_{11} -2= 0
x_{4} x_{12} -1= 0
x_{4} x_{14} +x_{3} x_{13} = 0
x_{11} x_{14} +x_{12} x_{13} = 0
x_{3} x_{6} +x_{4} x_{5} = 0
x_{5} x_{13} -6= 0
x_{6} x_{14} -6= 0
x_{7} x_{15} -10= 0
x_{8} x_{16} -6= 0
x_{8} x_{10} +x_{6} x_{9} = 0
x_{6} x_{12} +x_{5} x_{11} = 0
For the calculator:
(DynkinType =A^{2}_2+A^{28}_1; ElementsCartan =((2, 2, 3, 4, 3, 2), (0, -1, -1, -2, -2, -2), (0, 6, 6, 10, 6, 0)); generators =(x_{1} g_{-32}+x_{2} g_{-33}, x_{10} g_{33}+x_{9} g_{32}, x_{3} g_{21}+x_{4} g_{20}, x_{12} g_{-20}+x_{11} g_{-21}, x_{5} g_{-2}+x_{6} g_{-3}+x_{7} g_{-4}+x_{8} g_{-5}, x_{16} g_{5}+x_{15} g_{4}+x_{14} g_{3}+x_{13} g_{2}) );
FindOneSolutionSerreLikePolynomialSystem{}( x_{2} x_{10} +x_{1} x_{9} -2, x_{2} x_{10} +2x_{1} x_{9} -3, 2x_{2} x_{10} +x_{1} x_{9} -3, x_{2} x_{16} +x_{1} x_{14} , x_{9} x_{16} +x_{10} x_{14} , x_{1} x_{8} +x_{2} x_{6} , x_{3} x_{11} -1, x_{4} x_{12} +x_{3} x_{11} -2, x_{4} x_{12} -1, x_{4} x_{14} +x_{3} x_{13} , x_{11} x_{14} +x_{12} x_{13} , x_{3} x_{6} +x_{4} x_{5} , x_{5} x_{13} -6, x_{6} x_{14} -6, x_{7} x_{15} -10, x_{8} x_{16} -6, x_{8} x_{10} +x_{6} x_{9} , x_{6} x_{12} +x_{5} x_{11} )