Symmetries

By employing vector product rules (cf. wikipedia) prove that $A^*=-A$ and $A^2=-P$.

1. We use the formula: $\la b\times u,v\ra=\la u,v\times b\ra=-\la u,b\times v\ra$, which by definition of $A$ and $A^*$ is equivalent to $A^*=-A$.
2. The vector triple product formula (BAC-CAB formula) asserts that for all $a,b,c\in\R^3$: $a\times(b\times c)=b\la a,c\ra-c\la a,b\ra$. It follows that $A^2u=b\times(b\times u)=b\la b,u\ra-\norm b^2 u=b\la b,u\ra-u$ which by definition of $P$ coincides with $-Pu$.