Is there a name for this bilinear form which compares a linear operator with its adjoint?

Question

Let $X$ be a reflexive Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and let $A:X\to X$. Denote the adjoint of $A$ by $A^\ast$. Define the bilinear form $B:X\times X\to\mathbb R$ by $$ B(u,v)=\langle u,Av\rangle-\langle u,A^\ast v\rangle=\langle u,Av\rangle-\langle Au,v\rangle.$$ I use $B$ in an ad-hoc way to quantify how non-self-adjoint an operator is, with $B\equiv 0$ corresponding to a self-adjoint operator. It is particularly useful for seeing which terms in complicated operator lead to "non-self-adjointness". Informally, I have been calling $B$ the adjoint commutator, but I am curious if this bilinear form is used elsewhere and if it already has a commonly used name.