Is there an expression for $$\ln\left(e^{-i X}e^{iY}e^{iX}\right)$$ that is better than two nested campbell-baker-hausdorff series? To be specific, my interest is in the case of $Y$ being a general $su(6)$ element and $X$ being a general element of its $su(3)$ subalgebra.
This is a variation on Log of product of 3 matrix exponentials, where two of the exponents were totally equal.