Unitary similarity of algebras generated by pairs of orthoprojectors

It is shown that the unitary similarity of two matrix algebras generated by pairs of orthoprojectors $\{P_1,Q_1\}$ and $\{P_2,Q_2\}$ can be verified by comparing the traces of $P_1$, $Q_1$, and $(P_1Q_1)^i$, $i=1,2,\dots,n$, with those of $P_2$, $Q_2$, and $(P_2Q_2)^i$. The conditions of the unitary similarity of two matrices with quadratic minimal polynomials presented in [A. George and Kh. D. Ikramov, Unitary similarity of matrices with quadratic minimal polynomials. – Linear Algebra Appl., 349 (2002), 11–16] are refined.