Why is every derivation a Lie derivation and every Lie derivation is a Lie triple derivation?

Question

I’m studying Lie triple derivation of triangular algebra.. I know what is Lie derivation and what is Lie triple derivation but I am not getting the sense that why every derivation is a lie derivation and every lie derivation is a lie triple derivation ? But the converse is not true. Can you please help me in understanding this thing. Explaining with any relevant example will be highly appreciated.

Answer 1

A Lie triple derivation is by definition just a pre-derivation, i.e., a linear map $P\colon L\rightarrow L$ satisfying $$ P([x,[y,z]])=[P(x),[y,z]]+[x,[P(y),z]]+[x,[y,P(z)]]$$ for every $x,y,z\in L$. Of course, every ordinary derivation is a pre-derivation: Let $D\in Der(L)$. Then by definition $$D([x,[y,z]])=[x,D([y,z])]+[D(x),[y,z]]$$ Substituting $D([y,z])=[D(y),z]+[y,D(z)]$ we obtain $$D([x,[y,z]])=[x,[D(y),z]]+[x,[y,D(z)]]+[D(x),[y,z]].$$ The converse is not true in general. Take the Heisenberg Lie algebra.