I just derived this simple vector identity $$\mathbf{a}\cdot\left(\mathbf{b}\times\left(\mathbf{b} \times\mathbf{c}\right)\right) = \left(\mathbf{a}\times\mathbf{b}\right)\cdot\left(\mathbf{b}\times\mathbf{c}\right).$$
This can be done fairly easily:
Application of the vector triple product identity to the left hand side gives $$\mathbf{a}\cdot\left(\mathbf{b}\times\left(\mathbf{b} \times\mathbf{c}\right)\right) = \mathbf{a}\cdot\left((\mathbf{b}\cdot\mathbf{c})\mathbf{b}-(\mathbf{b}\cdot\mathbf{b})\mathbf{c}\right) =\mathbf{a}\cdot(\mathbf{b}\cdot\mathbf{c})\mathbf{b} -\mathbf{a}\cdot(\mathbf{b}\cdot\mathbf{b})\mathbf{c}.$$ Application of the scalar quadruple product identity to the right hand side gives $$\left(\mathbf{a}\times\mathbf{b}\right)\cdot\left(\mathbf{b}\times\mathbf{c}\right) = (\mathbf{a}\cdot\mathbf{b})(\mathbf{b}\cdot\mathbf{c})-(\mathbf{a}\cdot\mathbf{c})(\mathbf{b}\cdot\mathbf{b}).$$
From this, it directly follows that \begin{align} \mathbf{a}\cdot(\mathbf{b}\cdot\mathbf{c})\mathbf{b} &\stackrel{!}{=} (\mathbf{a}\cdot\mathbf{b})(\mathbf{b}\cdot\mathbf{c}) \text{ and}\\ \mathbf{a}\cdot(\mathbf{b}\cdot\mathbf{b})\mathbf{c} &\stackrel{!}{=} (\mathbf{a}\cdot\mathbf{c})(\mathbf{b}\cdot\mathbf{b}), \end{align} which is obviously a true statement.
I am currently writing a paper and would like to cite the formal name of this identity, but was unfortunately not able to find it anywhere online thus far. Does anybody have a clue what it is called in literature?
Thanks in advance!
The fact that $\left(\mathbf{a}\times\mathbf{b}\right)\cdot\left(\mathbf{c}\times\mathbf{d}\right) = (\mathbf{a}\cdot\mathbf{b})(\mathbf{c}\cdot\mathbf{d})-(\mathbf{a}\cdot\mathbf{c})(\mathbf{b}\cdot\mathbf{d})$ is sometimes called the Binet-Cauchy identity; an application of this amounts to a half of what you have proved.
Even on this page of identities, I didn't see anything like your expression on the left. I don't know what to call it since you've already used the term "scalar quadruple product".
Perhaps a good name for your identity would be a "scalar quadruple product identity" in the sense that it says that two such products are equal.