Make D, the differentiation operator, useable as a variable name

$\mathbf{local} \mathrm{D}≔\mathrm{D}\:$

$\mathrm{D}$$$

Define the vectors A, B, C, and D

$⟨3\,-2\,4⟩$$\stackrel{\text{assign to a name}}{\to }$$A$

$⟨2\,5\,-4⟩$$\stackrel{\text{assign to a name}}{\to }$$B$

$⟨5\,7\,6⟩$$\stackrel{\text{assign to a name}}{\to }$$C$

$⟨4\,3\,-2⟩$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{D}$

Evaluate the left-hand side of the identity

$\left(\mathbf{A}\times \mathbf{B}\right)\times \left(\mathbf{C}\times \mathbf{D}\right)$ = $\left[\begin{array}{c}\mathrm{-906}\\ \mathrm{-764}\\ 232\end{array}\right]$$$

Evaluate the right-hand side of the identity

$\left(\mathbf{A}·\left(\mathbf{C}\times \mathbf{D}\right)\right)\cdot \mathbf{B}-\left(\mathbf{B}·\left(\mathbf{C}\times \mathbf{D}\right)\right)\cdot \mathbf{A}$ = $\left[\begin{array}{c}\mathrm{-906}\\ \mathrm{-764}\\ 232\end{array}\right]$$$

Evaluate the left-hand side of the identity

$\left(\mathbf{A}\times \mathbf{B}\right)\times \left(\mathbf{C}\times \mathbf{D}\right)$ = $\left[\begin{array}{c}\mathrm{-906}\\ \mathrm{-764}\\ 232\end{array}\right]$$$

Evaluate the right-hand side of the identity

$\left(\mathbf{A}·\left(\mathbf{B}\times \mathbf{D}\right)\right)\mathbf{\cdot }\mathbf{C}\mathbf{-}\left(\mathbf{A}\mathbf{·}\left(\mathbf{B}\times \mathbf{C}\right)\right)\mathbf{\cdot }\mathbf{D}$ = $\left[\begin{array}{c}\mathrm{-906}\\ \mathrm{-764}\\ 232\end{array}\right]$$$

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{MultivariateCalculus}\right)\:$

$\mathbf{A}\,\mathbf{B}\,\mathbf{C}\,\mathbf{D}≔⟨3\,-2\,4⟩\,⟨2\,5\,-4⟩\,⟨5\,7\,6⟩\,⟨4\,3\,-2⟩\:$

Evaluate the left-hand side of the identity

$\mathrm{CrossProduct}\left(\mathrm{CrossProduct}\left(\mathbf{A}\,\mathbf{B}\right)\,\mathrm{CrossProduct}\left(\mathbf{C}\,\mathbf{D}\right)\right)$ = $\left[\begin{array}{c}\mathrm{-906}\\ \mathrm{-764}\\ 232\end{array}\right]$$$

Evaluate the right-hand side of the identity

$\mathrm{BoxProduct}\left(\mathbf{A}\,\mathbf{C}\,\mathbf{D}\right)\cdot \mathbf{B}-\mathrm{BoxProduct}\left(\mathbf{B}\,\mathbf{C}\,\mathbf{D}\right)\cdot \mathbf{A}$ = $\left[\begin{array}{c}\mathrm{-906}\\ \mathrm{-764}\\ 232\end{array}\right]$$$

Alternatively, use first principles to evaluate the right-hand side

$\mathrm{DotProduct}\left(\mathbf{A}\,\mathrm{CrossProduct}\left(\mathbf{C}\,\mathbf{D}\right)\right)\cdot \mathbf{B}-\mathrm{DotProduct}\left(\mathbf{B}\,\mathrm{CrossProduct}\left(\mathbf{C}\,\mathbf{D}\right)\right)\cdot \mathbf{A}$ = $\left[\begin{array}{c}\mathrm{-906}\\ \mathrm{-764}\\ 232\end{array}\right]$$$

Evaluate the left-hand side of the identity

$\mathrm{CrossProduct}\left(\mathrm{CrossProduct}\left(\mathbf{A}\,\mathbf{B}\right)\,\mathrm{CrossProduct}\left(\mathbf{C}\,\mathbf{D}\right)\right)$ = $\left[\begin{array}{c}\mathrm{-906}\\ \mathrm{-764}\\ 232\end{array}\right]$$$

Evaluate the right-hand side of the identity

$\mathrm{BoxProduct}\left(\mathbf{A}\,\mathbf{B}\,\mathbf{D}\right)\cdot \mathbf{C}-\mathrm{BoxProduct}\left(\mathbf{A}\,\mathbf{B}\,\mathbf{C}\right)\cdot \mathbf{D}$ = $\left[\begin{array}{c}\mathrm{-906}\\ \mathrm{-764}\\ 232\end{array}\right]$$$

Alternatively, use first principles to evaluate the right-hand side

$\mathrm{DotProduct}\left(\mathbf{A}\,\mathrm{CrossProduct}\left(\mathbf{B}\mathbf{\,}\mathbf{D}\right)\right)\mathbf{\cdot }\mathbf{C}\mathbf{-}\mathrm{DotProduct}\left(\mathbf{A}\,\mathrm{CrossProduct}\left(\mathbf{B}\mathbf{\,}\mathbf{C}\right)\right)\mathbf{\cdot }\mathbf{D}$ = $\left[\begin{array}{c}\mathrm{-906}\\ \mathrm{-764}\\ 232\end{array}\right]$$$