= Solution
In components, the <scalar triple product> is
$$
\mathbf a\cdot(\mathbf b\times\mathbf c)=a_1(b_2c_3-b_3c_2)+a_2(b_3c_1-b_1c_3)+a_3(b_1c_2-b_2c_1).
$$
It is the <determinant> with rows $\mathbf a,\mathbf b,\mathbf c$. A cyclic permutation of the rows has positive sign, so $\mathbf a\cdot(\mathbf b\times\mathbf c)=\mathbf c\cdot(\mathbf a\times\mathbf b)=(\mathbf a\times\mathbf b)\cdot\mathbf c$. Equivalently the <Levi-Civita symbol> expression $\epsilon_{ijk}a_ib_jc_k$ is unchanged by that cyclic relabeling.
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