Let $J$ be the largest ideal of Leibniz algebra $L$ which denotes the non-Lie character of $L$. Is it possible to write $L=L_{Lie}\cap J$? We know that $L_{Lie}= L/J$. I am going to give the following example:
Example: Let $L$ be a Leibniz algebra of dimension $2$, then we have four Leibniz algebras (up to isomorphism) as follows:
- Two Lie algebras: Abelian and Solvable given in suitable basis $a$ , $b$ by the relations $[a,b]=-[b,a]=b$.
- Two left Leibniz algebras: First is $[b,b]=a$ and the second one is $[b,a]=a$ , $[b,b]=a$.
What is the intersection between $L_{Lie}$ and $L$?
Definition:A (right) Leibniz algebra, named after Gottfried Wilhelm Leibniz, sometimes called a Loday algebra, after Jean-Louis Loday, is a module L over a commutative ring R with a bilinear product [ _ , _ ] satisfying the Leibniz identity
$$[[a,b],c] = [a,[b,c]]+ [[a,c],b].$$