The value of 

$|\mathbf{a}\mathbf{ }\times (\mathbf{i}\times \mathbf{j}){|}^2+ |\mathbf{a}\mathbf{ }\times (\mathbf{j}\times \mathbf{k}){|}^2+ |\mathbf{a}\mathbf{ }\times ( \mathbf{k}\times \mathbf{i}){|}^2$ is

For $\mathbf{a}= a_1\mathbf{i}\mathbf{ }+ a_2\mathbf{j}\mathbf{ }+ a_3\mathbf{k}$, we have

$\begin{array}{l}\mathbf{a}\mathbf{ }\times (\mathbf{i}\mathbf{ }\times \mathbf{j})=(\mathbf{a}\cdot \mathbf{i})\mathbf{j}-( \mathbf{a}\cdot \mathbf{j}) \mathbf{i}\\ \mathbf{a}\mathbf{ }\times (\mathbf{i}\mathbf{ }\times \mathbf{j}{)}^2=(\mathbf{a}\cdot \mathbf{i}{)}^2+ (\mathbf{a}\cdot \mathbf{j}{)}^2=a_1^2+a_2^2\\ \mathbf{a}\times {\left.(\mathbf{j}\mathbf{ }\times \mathbf{k})\right|}^2=a_2^2+a_3^2\\ \mathbf{a}\times {\left.(\mathbf{k}\mathbf{ }\times \mathbf{i})\right|}^2=a_3^2+a_1^2\end{array}$

so the required value  $=2\left(a_1^2 + a_2^2 + a_3^2\right)=2|\mathbf{a}{|}^2$