Algebra for the Baby

I am reading the following article.

Ryba, Alexander J.E., A natural invariant algebra for the Baby Monster group., J. Group Theory 10, No. 1, 55-69 (2007). ZBL1228.20012..

Author works with 4370-dimensional representation of Baby Monster group over field with two elements. The formula for multiplication in the 4370-dimensional vector space is following:

$\underline t*\underline{t'}=\underline{t'}(t+1) + (\underline{t'},\underline{t})\underline{t} + \lambda_t(\underline{t'})$

Symbol $\underline t$ denotes axis of transposition $t$ in $B$. The transposition is $2A$ involution. The "axis of transposition" is the unique vector preserved by the centralizer of the transposition. There is prove that this multiplication is commutative and it makes vector space the algebra. We have also $x^2=0$ for each algebra element. In last chapter there is classification of subalgebras generated by two axes. Result depends on conjugacy class of product of transpositions.

The drawback of this formula is $\lambda_t$ which is constructed by use of representation theory. I would like to see a direct formula for multiplication within the algebra, in order to see the symmetries of baby monster. I would like to obtain the algebra in GAP, in which I can test some hypotheses about the algebra.

I have the following questions related to this algebra.

1. Let $L'$ denote $78$-dimensional simple Lie algebra generated by GAP command SimpleLieAlgebra("E", 6, GF(2)). From Andrei Smolensky answer in the comment I conclude that automorphism group of this Lie algebra is $E_6(2)$. This algebra has 12 generators: basis elements 1..6 and 37..42. How to construct Lie algebra of dimension 78 over $\mathbb F_2$ mentioned in the paper ? It's automorphism group should be $^2E_6(2)$. Should I start from GAP $E_6$ over $\mathbb F_4$ ? I would like to have explicit construction of subalgebra $L_t$ of algebra $V_{4370}$ for a given transposition $t$ (see section 3 for the definition of $L$ and section 4 for the definition of $L_t$ in the paper).
2. There is a $782$-dimensional subalgebra invariant for the $Fi_{23}$ subgroup. This subgroup does not have any axis fixed. This subalgebra is spanned by $31671$ axes of conjugacy class $2A$ of $Fi_{23}$. Using the information from chapter 6 in this paper about "dihedral subalgebras" we could define such an algebra, I believe. The multiplication in this subalgebra is defined by $\underline t*\underline u=\underline t+\underline u+\underline{t^u}$ for not commuting $t,u$ otherwise it is $0$.
3. I am not sure how to define the $5$-dimensional dihedral subalgebra for the 4B case, because I don't follow how $l_t$ is defined in this paper.
4. From direct calculations of eigenspaces in GAP I observed that a given subgroup $H$ of $B$ has vectors fixed only when the stabilizer of $H$ is not trivial. This is experimental data, I don't know how to prove it.

I posted the same question on math.stackexchange three weeks ago with no answer there, so I try it here.

Mnemonic way to remember dimension $4371$ is to exchange middle digits and multiply by $3$: $3*47*31=4371$.

Side note: Most of the authors spell "Baby Monster group" not "baby monster group". All sporadic groups except monster and baby monster are named from someones name. I wonder whether we should use capital letters for "baby monster group" ? In this case "monster" is rather kind of animal, isn't it ?

• For question 4, which action are you using? – Watson Ladd Dec 12 '17 at 20:01
• By "stabilizer of H" I mean set of elements in B which commute with H. For example subgroup $2.^2E_6(2).2$ or.second maximal have non trivial stabilizer, while $Fi_{23}$ or $Th$ have trivial one. – Marek Mitros Dec 12 '17 at 21:41
• But what about the vectors fixed by it? – Watson Ladd Dec 12 '17 at 23:22
• I use the normal action matrix times vector - the same which is considered in the paper. I used Eigenspaces and Eigenvectors functions from GAP to determine vector or subspace fixed by given subgroup H. I can show you the GAP code I used. For example for first max it is enough to take two random elements of order 70 from it. Intersection of eigenspaces give axis of transposition. – Marek Mitros Dec 13 '17 at 6:40
• On your first question: the automorphism group is the adjoint group of type $E_6$ (this is, in fact, its definition). Since there is no nontrivial involution on $GF(2)$, the group ${}^2E_6$ is no other than $F_4$. As for the generators, you are correct, 12 generators suffice, they correspond to the fundamental roots of its root system, see Serre's theorem on the generators and relation for semisimple Lie algebras. – Andrei Smolensky Dec 13 '17 at 13:18