Does the following identity hold: $ [{A \times B^* + A^* \times B} ]$ = $2Re{[A \times B^*]}$

Question

This seems to be true at first glance following that $a + a^*$ = $2Re(a)$

In any case, can someone help me verify whether this identity holds?

Note: $\times$ is the cross product

Answer 1

The LHS is

$$A \times B^* + A^* \times B=\frac12(A^*B^*+AB) +\frac12(AB+A^*B^*)=AB+A^*B^*.$$

The RHS is

$$2\mathrm{Re} (A \times B^*)=\mathrm{Re}(A^*B^*+AB)=\frac{A^*B^*+AB+(A^*B^*+AB)^*}{2}\\=\frac{A^*B^*+AB+AB+A^*B^*}{2}=AB+A^*B^*.$$

So, the equality is an identity.