Let $A$ be a finite-dimensional (not necessarily unital) associative algebra over the field of complex numbers $\mathbb{C}$ (but I'm also interested in more general fields). Assume the multiplication on $A$ is non-degenerate, which means that $A= AA$ and if $a \in A$ satisfies $aA = 0$ or $Aa = 0$, then $a=0$. Is it true that $A$ is unital? If not, what is a counterexample?

Some easy observations:

If $A$ is also simple, then it can be shown that the answer is positive. This follows for instance by the argument here.

If $A$ is a $C^*$-algebra (which is automatically non-degenerate), then a finite-dimensional $C^*$-algebra is automatically unital.