Show that the group operation of conjugation is compatible with the bilinear form $\langle A,B\rangle=1/2 \textrm{ trace }AB$

Question

Question 6.6 of General linear groups (Artin Algebra):

(a) Show that $O_2$ operates by conjugation on its Lie Algebra

(b) Show that this operation is compatible with the bilinear form $\langle A, B\rangle = 1/2 \textrm{ trace }AB. $

(c) Use the operation to define a homomorphism $O_2 \rightarrow O_2$, and describe this homomorphism explicitly.

(a) The Lie Algebra of $O_2$ is $2\times 2$ skew-symmetric matrices and hence we have to show that given $P\in O_2$ and $Q$ a skew-symmetric matrix, $P^{-1}QP$ is skew-symmetric and that the operation is associative; both are easy to verify.

(b) I do not understand what "compatible" means. My first instinct is that to show $\langle A, B\rangle=\langle P^{-1}AP, P^{-1}BP\rangle$.

(c) I do not see how one can define an homomorphism from $O_2$ to itself with the knowledge of (a) and (b)

I request you to help me with (b) and (c). Please provide hints in place of answers wherever possible. Thank you