Answer
Using the properties:
${\bf{u}}\cdot{\bf{w}} = {\bf{w}}\cdot{\bf{u}}$, ${\ }$ ${\bf{u}}\cdot{\bf{v}} = {\bf{v}}\cdot{\bf{u}}$, and
${\bf{v}}\cdot{\bf{w}} = {\bf{w}}\cdot{\bf{v}}$,
we obtain
${\bf{u}} \times \left( {{\bf{v}} \times {\bf{w}}} \right) + {\bf{v}} \times \left( {{\bf{w}} \times {\bf{u}}} \right) + {\bf{w}} \times \left( {{\bf{u}} \times {\bf{v}}} \right) = {\bf{0}}$
Work Step by Step
We have the identity:
(1) ${\ \ }$ ${\bf{u}} \times \left( {{\bf{v}} \times {\bf{w}}} \right) = \left( {{\bf{u}}\cdot{\bf{w}}} \right){\bf{v}} - \left( {{\bf{u}}\cdot{\bf{v}}} \right){\bf{w}}$
By exchanging of variables, we obtain
(2) ${\ \ }$ ${\bf{v}} \times \left( {{\bf{w}} \times {\bf{u}}} \right) = \left( {{\bf{v}}\cdot{\bf{u}}} \right){\bf{w}} - \left( {{\bf{v}}\cdot{\bf{w}}} \right){\bf{u}}$
(3) ${\ \ }$ ${\bf{w}} \times \left( {{\bf{u}} \times {\bf{v}}} \right) = \left( {{\bf{w}}\cdot{\bf{v}}} \right){\bf{u}} - \left( {{\bf{w}}\cdot{\bf{u}}} \right){\bf{v}}$
Summing equations (1), (2), and (3) gives
${\bf{u}} \times \left( {{\bf{v}} \times {\bf{w}}} \right) + {\bf{v}} \times \left( {{\bf{w}} \times {\bf{u}}} \right) + {\bf{w}} \times \left( {{\bf{u}} \times {\bf{v}}} \right)$
$ = \left( {{\bf{u}}\cdot{\bf{w}}} \right){\bf{v}} - \left( {{\bf{u}}\cdot{\bf{v}}} \right){\bf{w}}$
$ + \left( {{\bf{v}}\cdot{\bf{u}}} \right){\bf{w}} - \left( {{\bf{v}}\cdot{\bf{w}}} \right){\bf{u}}$
$ + \left( {{\bf{w}}\cdot{\bf{v}}} \right){\bf{u}} - \left( {{\bf{w}}\cdot{\bf{u}}} \right){\bf{v}}$
Since ${\bf{u}}\cdot{\bf{w}} = {\bf{w}}\cdot{\bf{u}}$, ${\ }$ ${\bf{u}}\cdot{\bf{v}} = {\bf{v}}\cdot{\bf{u}}$, and ${\bf{v}}\cdot{\bf{w}} = {\bf{w}}\cdot{\bf{v}}$, therefore
${\bf{u}} \times \left( {{\bf{v}} \times {\bf{w}}} \right) + {\bf{v}} \times \left( {{\bf{w}} \times {\bf{u}}} \right) + {\bf{w}} \times \left( {{\bf{u}} \times {\bf{v}}} \right) = {\bf{0}}$
