Proof that the cross product is not associative without using components
Consider two non-zero perpendicular vectors $\def\v#1{{\bf#1}}\v a$ and $\v b$. We have $$(\v a \times\v a)\times\v b=\v0\times\v b=\v0\ .$$ However $\v a\times\v b$ is perpendicular to $\v a$, and is not the zero vector, so $$\v a\times(\v a\times \v b)\ne\v 0\ .$$ Therefore $$(\v a \times\v a)\times\v b\ne\v a\times(\v a\times \v b)\ .$$