Fixed subalgebra of a commutative Frobenius algebra

Let $B$ be a finite-dimensional commutative algebra generated by a single element, and let $A = B \otimes B$. We prove that the fixed subalgebra of $A$ under the involution ${b_1} \otimes {b_2} \mapsto {b_2} \otimes {b_1}$ is Frobenius if and only if either the characteristic of $B$ is different from 2 or $B$ is separable.