Putnam 2010 A5

Let $G$ be a group, with operation $*$. Suppose that

(i) $G$ is a subset of $\mathbb{R}^3$ (but $*$ need not be related to addition of vectors);

(ii) For each $\mathbf{a},\mathbf{b} \in G$, either $\mathbf{a}\times \mathbf{b} = \mathbf{a}*\mathbf{b}$ or $\mathbf{a}\times \mathbf{b} = 0$ (or both), where $\times$ is the usual cross product in $\mathbb{R}^3$.

Prove that $\mathbf{a} \times \mathbf{b} = 0$ for all $\mathbf{a}, \mathbf{b} \in G$.