By the vector triple product identity: $\mathbf{a}\times(\mathbf{b}\times\mathbf{c}) = (\mathbf{a}\cdot\mathbf{c})\mathbf{b} - (\mathbf{a}\cdot\mathbf{b})\mathbf{c}$, which is a linear combination of $\mathbf{b}$ and $\mathbf{c}$ — lying in the plane of $\mathbf{b}$ and $\mathbf{c}$. Similarly $\mathbf{a}\times(\mathbf{b}\times\mathbf{c})$ lies in the plane of $\mathbf{b}$ and $\mathbf{c}$. For the plane of $\mathbf{a}$ and $\mathbf{b}$, we need $(\mathbf{b}\times\mathbf{c})\times\mathbf{a}$ or equivalent.