I want to prove that:
if $L$ is leibniz algebra and $J(L)$is jacobson radical of $L$ always: $$J(L)\subset L^2$$.
I read this proof in (CLASSIFYING SEVERAL CLASSES OF LEIBNIZ ALGEBRAS Batten Ray)
this prove is:


if $x$ is not in $L^2$ ,then we can find a complementary subspace ,$M$ ,of $x$ in $L$ that contains $L^2$and since $L^2\subset M$,$M$ is maximal ideal(?) of $L$ and $x$ is not in $M$.
I dont know why $M$ is maximal ideal?


thanks for help



jacobson radical and leibniz algebra
Since it is a complementary subspace it has codimension one in $L$, and so is maximal. It is an ideal of $L$ because $L^2$ is inside $M$.


