Let $A$ be a nilpotent square matrix, $J$ be the antidiagonal matrix with 1's on the secondary diagonal (i.e. $J^{2}=E$) and let $B=J A J.$ Suppose we conjugate the matrices $A,B$ by a matrix $C:$ $A_1=C A C^{-1}, B_1=C B C^{-1}.$

**Question 1.** Suppose we know the matrix $A_1$ only (but don't know matrices $A$ and $C.$) Is there any trick (i.e. a $C$-invariant construction) to find the matrix $B_1$?

**Question 2.** Is it possible (knowing the matrix $A_1$ only) to find the conjugated commutator $C[A,B]C^{-1}$?