Let us say that $A$ is a (finite-dimensional) algebra over a field of characteristic zero. We can assume commutativity but not associativity, if that makes it easier. Indeed, I am mostly interested in the case of complex Jordan algebras.

Question: What is known about left-multiplication operators $L_a:A\to A$, $L_ax=ax$, that are derivations of $A$, in the sense that $L_a(xy)=(L_ax)y+x(L_ay)$ for all $x$, $y\in A$? What about algebras such that all left-multiplications are derivations?

I think these are never semisimple algebras. An obvious remark is that $A^3=0$ is a sufficient condition for $L(A)\subset\mathrm{Der}(A)$.

(Of course, Lie algebras fit that category.)