Answer
Using Eq. (5) we obtain
$\left( {{\bf{i}} \times {\bf{j}}} \right) \times {\bf{j}} \ne {\bf{i}} \times \left( {{\bf{j}} \times {\bf{j}}} \right)$
Work Step by Step
From Eq. (5) we have
${\bf{i}} \times {\bf{j}} = {\bf{k}}$, ${\ \ }$ ${\bf{j}} \times {\bf{k}} = {\bf{i}}$, ${\ \ }$ ${\bf{k}} \times {\bf{i}} = {\bf{j}}$
${\bf{i}} \times {\bf{i}} = {\bf{j}} \times {\bf{j}} = {\bf{k}} \times {\bf{k}} = {\bf{0}}$
So,
$\left( {{\bf{i}} \times {\bf{j}}} \right) \times {\bf{j}} = {\bf{k}} \times {\bf{j}} = - {\bf{i}}$
${\bf{i}} \times \left( {{\bf{j}} \times {\bf{j}}} \right) = {\bf{i}} \times {\bf{0}} = {\bf{0}}$
Therefore, $\left( {{\bf{i}} \times {\bf{j}}} \right) \times {\bf{j}} \ne {\bf{i}} \times \left( {{\bf{j}} \times {\bf{j}}} \right)$. We conclude that the Associative Law does not hold for cross products.
