Let the bilinear trace form of a finite-dimensional associative algebra be defined as:

$(u,v) \mapsto Tr(L_u L_v)$

For $L_u$ the linear map given by multiplication on the left by $u$. In the literature, there seem to be good characterizations of algebras where this form is non-degenerate (semi-simple, special Frobenius), but what about the other extreme?

Is there a characterization for an associative algebra $A$ whose bilinear trace form, considered as a linear map $V \to V^*$ is rank 1?

In particular, when $A$ is unital and commutative, there seems to be only one possibility for the trace form, if its rank 1:

$(u,v)\mapsto \frac{1}{dimV} Tr(L_u)Tr(L_v)$

This seems like quite the coincidence. Could anyone shed some light on this?