Is the following equality is true for trace?
$\mathrm{Re}\bigl(\operatorname{Tr}(-YZ^*X+YXZ^*)\bigr)=\mathrm{Re} \bigl(\operatorname{Tr}(XYZ-YXZ)\bigr)$ for complex matrices $X,Y,Z.$
Actually I am trying to show that with respect to certain inner product on the Lie algebra of a linear connected semisimple Lie group $-\mathrm{ad}(X^*)=(\mathrm{ad}X)^*$.