I want to show how can be expressed the following difference involving triple products: given four vectors $\mathbf {a}, \mathbf b, \mathbf c, \mathbf d \in \mathbb R^3$ \begin{equation} [(\mathbf a \times \mathbf b) \cdot \mathbf c] \, \mathbf d - (\mathbf a \cdot \mathbf c) (\mathbf b \times \mathbf d) \end{equation}
This because in my application the second expression involving the term $\mathbf b \times \mathbf d$ has physical meaning. I have played with the properties of the triples product without getting results. I cannot find an expression that connects the two quantities.