frattini ideal of leibniz algebras

Question

Let $L$ be a leibniz algebra over $K$, an algebraically closed field of characteristic $0$.

if $L=R+S$, where $R$ is solvable ideal, $S\simeq sl_2(K)\oplus...\oplus sl_2(K)$.why $\Phi(L)$ is nilpotent ideal in $L$?

$\Phi(L)$ is largest ideal of $L$ contained in Frattini subalgebr of $L$

thanks for help

Answer 1

The Frattini ideal $\Phi(L)$ of a finite-dimensional Leibniz algebra $L$ of characteristic zero is, as in the Lie algebra case, always nilpotent, see for example Theorem $5.5$ and its Corollary $5.6$ in the article Some Theorems on Leibniz Algebras by Donald W. Barnes. In case $L$ is elementary, i.e., a direct sum of $\mathfrak{sl}(2)$'s and solvable algebras with basis $(e_i,f_i)$ and products $e_if_j=\lambda_{ij}f_j$, then $\Phi(L)=0$.