The vector $\vec{a}\times (\vec{b}\times \vec{a})$ is coplanar with:

Using the vector triple product formula:
$$\vec{a}\times (\vec{b}\times \vec{c})=(\vec{a}\cdot \vec{c})\vec{b}-(\vec{a}\cdot \vec{b})\vec{c}$$
Substituting $\vec{c}=\vec{a}$:
$$\vec{a}\times (\vec{b}\times \vec{a})=(\vec{a}\cdot \vec{a})\vec{b}-(\vec{a}\cdot \vec{b})\vec{a}$$
The resulting vector is a linear combination of $\vec{a}$ and $\vec{b}$. Therefore, it lies in the plane containing both vectors $\vec{a}$ and $\vec{b}$.