The value of $\hat{\mathbf{i}}\times (\hat{\mathbf{i}}\times \mathbf{a})+\hat{\mathbf{j}}\times (\hat{\mathbf{j}}\times \mathbf{a})+\hat{\mathbf{k}}\times (\hat{\mathbf{k}}\times \mathbf{a})$ is

Let $\mathbf{a}=a_1\hat{\mathbf{i}}+a_2\hat{\mathbf{j}}+a_3\hat{\mathbf{k}}$, Then,

$\hat{\mathbf{i}}\times \mathbf{a}=\hat{\mathbf{i}}\times \left(a_1\hat{\mathbf{i}}+a_2\hat{\mathbf{j}}+a_3\hat{\mathbf{k}}\right)$

$=a_2\hat{\mathbf{k}}-a_3\hat{\mathbf{j}}$

$\hat{\mathbf{j}}\times \mathbf{a}=\hat{\mathbf{j}}\times \left(a_1\hat{\mathbf{i}}+a_2\hat{\mathbf{j}}+a_3\hat{\mathbf{k}}\right)$

$=-a_1\hat{\mathbf{k}}+a_3\hat{\mathbf{i}}=a_3\hat{\mathbf{i}}-a_1\hat{\mathbf{k}}$

$\hat{\mathbf{k}}\times \mathbf{a}=\hat{\mathbf{k}}\times \left(a_1\hat{\mathbf{i}}+a_2\hat{\mathbf{j}}+a_3\hat{\mathbf{k}}\right)$

$=a_1\hat{\mathbf{j}}-a_2\hat{\mathbf{i}}$

Now, $\hat{i}\times (\hat{i}\times \mathbf{a})+\hat{j}\times (\hat{j}\times \mathbf{a})+\hat{k}\times (\hat{k}\times \mathbf{a})$

$=\hat{\mathbf{i}}\times \left(a_2\hat{\mathbf{k}}-a_3\hat{\mathbf{j}}\right)+\hat{\mathbf{j}}\times \left(a_3\hat{\mathbf{i}}-a_1\hat{\mathbf{k}}\right)+\hat{\mathbf{k}}\times \left(a_1\hat{\mathbf{j}}-a_2\hat{\mathbf{i}}\right)$

$=-a_2\hat{j}-a_3\hat{k}-a_3\hat{k}-a_1\hat{i}-a_1\hat{i}-a_2\hat{j}$

$=-2\left(a_1\hat{\mathbf{i}}+a_2\hat{\mathbf{j}}+a_3\hat{\mathbf{k}}\right)$

$=-2\overrightarrow{\mathrm{a}}$