I want to calculate matrix multiplication $ ABA $, where $ A $ Y $ B $ They are real and orthogonal matrices. In fact, they are specifically $ 3 times3 $ rotation matrices However, it is much easier if I can reverse the order of $ BA $ somehow, because I can perform multiplication much more easily.

I know that matrix multiplication is not commutative, however, I ask why both $ A $ Y $ B $ they are orthogonal matrices and, hopefully, there may be some trick to use their orthogonality to reorder the product.

I tried to solve this, but I got stuck here:

$$

ABA = A ((BA) -1) -1 – = A (A -1 B -1) -1

$$

Is there any way to proceed from here?