Consider an antilinear involution, that is an antilinear map on a complex vector space, whose matrix $M$ obeys $MM^*=1$ where the star denotes complex conjugation. Can we find a change of basis whose matrix $\Lambda$ would be such that $\Lambda^* M \Lambda^{-1} = 1$?

By taking the real components of $M$ and $\Lambda$, this can be reduced to a special case of the following problem: given four real commuting square matrices of the same size $A,B,C,D$ such that $AD=BC$, do real vectors $X$ and $Y$ such that $AX=BY$ and $CX=DY$ span the whole space?

The motivation for this question comes from quantum mechanics, where Hermitian conjugation is a antilinear involution on the space of operators. Trivializing this involution means finding a Hermitian basis of operators.