Consider a matrix Lie group equipped with a left &/or right invariant metric.
The adjoint of linear transformation $A$ with respect to the inner product is denoted as $A^*$.
Here what is actually meant by $A^*$? How to compute an expression like $(ad(X))^*(Y)$ where X,Y are lie algebra elements.
Note the definition: $ < (ad(X))^*(Y) , Z > = < Y, [X,Z] >$ for any lie algebra element $Z$.