Graded Lie algebras with zero component equal to a sum of commuting ideals

This paper considers transitive irreducible 1-graded Lie algebras $L=\bigoplus_{i\geqslant-1}L_i$, $L_1=0$, over an algebraically closed field $K$ of characteristic $p\geqslant0$, $p\ne2$. We prove that if $L_0=G_1+\dots+G_s$, $G_i\ne Z(L_0)$, is the decomposition of $L_0$ and the ideals of $G_i$ commute, then $s=1$ or $s=2$. In the latter case $L$ is isomorphic to one of the algebras $A_n$, $A^z_{n_0p-1}$ or $\widetilde{\mathfrak{gl}}(n_0p)=\mathfrak{gl}(n_0p)/\langle1\rangle$. .
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