Examples of three dimensional Non Nilpotent Leibniz or Lie algebras.

Question

Can one give me some examples of three dimensional Non Nilpotent Leibniz algebras? Any references to the classification of three dimensional Non Nilpotent Leibniz or Lie algebras will also be helpful.

Answer 1

Just for Lie algebras: $\mathfrak{sl}_2(\Bbb C)$ (over $\Bbb C$). $\mathfrak{su}_2$ (over $\Bbb R$). The upper triangular matrices in $\mathfrak{gl}_2(k)$ (over your favourite field $k$).

Answer 2

The classification of all complex $3$-dimensional Leibniz algebras is well-known, see for example here. Like for Lie algebras, there is a version of Engel's theorem to decide whether or not a given Leibniz algebra is nilpotent - see here. In fact, $L$ is nilpotent if and only if all left multiplication operators $d_a$ for $a\in L$ are nilpotent. So we can easily find out the non-nilpotent algebras from the classification list.