It is well know that any analytic function of an $n \times n$ real/complex matrix $f(A)$ can be expressed as linear combination of the first $n$ powers of $A$ by the Cayley-Hamilton theorem.
Is it possible to express the function (adjoint action of $SU(4)$ on $\mathfrak{su}(4)$ in a similar way. I.e. is it possible to write:
$UAU^{\dagger} = aI + bA + cA^2 + dA^3$.
and actually find formulas for $a,b,c,d$ in terms of $U$ and $A$?
Would it be possible in the case that $U=\exp(B)$ for some $B\in \mathfrak{su}(4)$ ?
If this were possible, $UAU^\dagger$ would always commute with $A$.