Let $\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}}\text{ and }\overrightarrow{\mathrm{c}}$ be three non-coplanar vectors and $\overrightarrow{r}$ be any arbitrary vector. Then $(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})\times (\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{c}})+(\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}})\times (\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{a}})+(\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{a}})\times (\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{b}})$ i, always equal to

$\begin{array}{l}(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})\times (\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{c}})\\ =\left(\right(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})\cdot \overrightarrow{\mathrm{c}})\overrightarrow{\mathrm{r}}-\left(\right(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})\cdot \overrightarrow{\mathrm{r}})\cdot \overrightarrow{\mathrm{c}}\\ =\left[\overrightarrow{\mathrm{a}}\overrightarrow{\mathrm{b}}\overrightarrow{\mathrm{c}}\right]\overrightarrow{\mathrm{r}}-\left[\overrightarrow{\mathrm{a}}\overrightarrow{\mathrm{b}}\overrightarrow{\mathrm{r}}\right]\overrightarrow{\mathrm{c}}\\ \text{ Similarly, }(\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}})\times (\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{a}})\\ =\left[\overrightarrow{\mathrm{b}}\overrightarrow{\mathrm{c}}\overrightarrow{\mathrm{a}}\right]\overrightarrow{\mathrm{r}}-\left[\overrightarrow{\mathrm{b}}\overrightarrow{\mathrm{c}}\overrightarrow{\mathrm{r}}\right]\overrightarrow{\mathrm{a}}\\ \text{ and }(\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{a}})\times (\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{b}})=\left[\overrightarrow{\mathrm{c}}\overrightarrow{\mathrm{a}}\overrightarrow{\mathrm{b}}\right]\overrightarrow{\mathrm{r}}-\left[\overrightarrow{\mathrm{c}}\overrightarrow{\mathrm{a}}\overrightarrow{\mathrm{r}}\right]\overrightarrow{\mathrm{b}}\\ \Rightarrow (\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})\times (\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{c}})+(\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}})\times (\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{a}})+(\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{a}})\times (\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{b}})\\ =3\left[\overrightarrow{\mathrm{a}}\overrightarrow{\mathrm{b}}\overrightarrow{\mathrm{c}}\right]\overrightarrow{\mathrm{r}}-\left(\right[\overrightarrow{\mathrm{b}}\overrightarrow{\mathrm{c}}\overrightarrow{\mathrm{r}}]\overrightarrow{\mathrm{a}}+[\overrightarrow{\mathrm{c}}\overrightarrow{\mathrm{a}}\overrightarrow{\mathrm{r}}]\overrightarrow{\mathrm{b}}+[\overrightarrow{\mathrm{a}}\overrightarrow{\mathrm{b}}\overrightarrow{\mathrm{r}}\left]\overrightarrow{\mathrm{c}}\right)\\ =3\left[\overrightarrow{\mathrm{a}}\overrightarrow{\mathrm{b}}\overrightarrow{\mathrm{c}}\right]\overrightarrow{\mathrm{r}}-\left[\overrightarrow{\mathrm{a}}\overrightarrow{\mathrm{b}}\overrightarrow{\mathrm{c}}\right]\overrightarrow{\mathrm{r}}=2\left[\overrightarrow{\mathrm{a}}\overrightarrow{\mathrm{b}}\overrightarrow{\mathrm{c}}\right]\overrightarrow{\mathrm{r}}\end{array}$