Answer
Let $\left\{ {{\bf{v}},{\bf{e}},{\bf{v}} \times {\bf{e}}} \right\}$ forms a right-handed system as is shown in the figure. It is proved from the diagram that ${\bf{e}} \times \left( {{\bf{v}} \times {\bf{e}}} \right) = \left( {{\bf{e}} \times {\bf{v}}} \right) \times {\bf{e}} = {\bf{v}}$
Work Step by Step
Suppose ${\bf{e}}$ is a unit vector orthogonal to ${\bf{v}}$. Let $\left\{ {{\bf{v}},{\bf{e}},{\bf{v}} \times {\bf{e}}} \right\}$ forms a right-handed system as is shown in the figure. Then, it follows from the right-hand rule, ${\bf{e}} \times \left( {{\bf{v}} \times {\bf{e}}} \right) = {\bf{v}}$. Also from the right-hand rule, $\left( {{\bf{e}} \times {\bf{v}}} \right) \times {\bf{e}} = {\bf{v}}$ as is shown in the diagram. Therefore, ${\bf{e}} \times \left( {{\bf{v}} \times {\bf{e}}} \right) = \left( {{\bf{e}} \times {\bf{v}}} \right) \times {\bf{e}} = {\bf{v}}$.