Let $\overrightarrow A \times \overrightarrow B = \overrightarrow C$
The cross product of $\overrightarrow B$ and $\overrightarrow A$ is perpendicular to the plane containing $\overrightarrow A$ and $\overrightarrow B$
i.e. perpendicular to $\overrightarrow B$

If a dot product of this cross product and $\overrightarrow A$ is taken, as
the cross product is perpendicular to $\overrightarrow A$, $\overrightarrow C$ $\times$ $\overrightarrow A$ = 0

Therefore product of $\left( {\overrightarrow A \times \overrightarrow B } \right) \times \overrightarrow A = 0$