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.)



Left- (right-) multiplications of an algebra that are derivations
An algebra whose (left) multiplications are derivations is referred to as a (left) *Leibniz algebra* (or *Loday algebra*). There is a large literature about this class of non-associative algebras. See e.g. the following survey by Joerg Feldvoss: <https://arxiv.org/abs/1802.07219>.


