On lie algebras with monomial basis

A basis $D$ of an algebra $L$ over a field $\Phi$ is called monomial if $ab=\alpha_{ab}c$ for all $a,b,c\in D$ and $\alpha_{ab}\in\Phi$. Such a basis is said to be homogeneous if $\alpha_{ab}\in\{-1,0,1\}$. A subalgebra $S$ in $L$ generated by elements of $D$ is called a $D$-subalgebra and the minimal number of generators for $S$ that belong to $D$ is called,the rank of $S$. We study Lie algebras with a monomial basis $D$ such that every pair of its elements generates a subalgebra in $L$ which is abelian or 3-dimensional simple.
All connected algebras of rank 3 are listed: they are an algebra of type $D_2$ over an arbitrary field, a 7-dimensional simple algebra of characteristic 3, and two families of 7-dimensional simple algebras of characteristic 2 (Theorem 2.1).
In the case when $L$ includes no 7-dimensional simple $D$-subalgebras, we prove that $D$ is embeddable as a set of 3-transpositions into some group $G$ and, moreover, the multiplication in $L$ is determined by the group multiplication to within structure constants. This, in particular, shows that the algebra $L$ is locally finite.
In the case when $G$ is a symmetric group $\Sigma_{\Omega}$, simple formulas for the multiplication in $L$ are found. Furthermore, if $|\Omega|=m<\infty$ then $L$ is an algebra of type $D_n$ for $m=2n$, and a classical algebra of type $B_n$ for $m=2n+1$.