Text: Differential Geometry, by Erwin Kreyszig
Given a set of vectors: $$ \overrightarrow{a}=\langle a_1, a_2, a_3 \rangle $$ $$ \overrightarrow{b}=\langle b_1, b_2, b_3 \rangle $$ $$ \overrightarrow{c}=\langle c_1, c_2, c_3 \rangle $$ undergoing a direct congruent transformation: $$ \bar{z_i}=\sum_{k=1}^3 \epsilon_{ik}z_k \qquad \sum_{i=0}^3 \epsilon_{ik} \epsilon_{il} = \delta_{kl}=\begin{cases} 0 & \text (k\neq l) \\ 1 & \text (k = l) \end{cases} \qquad \det(\epsilon_{ik})=1 \qquad (i=1,2,3)$$ mapping $\mathbb R^3 \mapsto \mathbb R^3 $, we need to prove the invariance of the mixed product:
$$ \overrightarrow{a} \cdot(\overrightarrow{b}\times\overrightarrow{c})=\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \\ \end{vmatrix}=\begin{vmatrix} \bar{a_1} & \bar{b_1} & \bar{c_1} \\ \bar{a_2} & \bar{b_2} & \bar{c_2} \\ \bar{a_3} & \bar{b_3} & \bar{c_3} \\ \end{vmatrix}=\bar{a} \cdot(\bar{b}\times\bar{c})$$
Proof. Beginning by substituting the given definition for the transformation into the determinant, we have
$$ \bar{a} \cdot(\bar{b}\times\bar{c})=\begin{vmatrix} \bar{a_1} & \bar{b_1} & \bar{c_1} \\ \bar{a_2} & \bar{b_2} & \bar{c_2} \\ \bar{a_3} & \bar{b_3} & \bar{c_3} \\ \end{vmatrix}=\begin{vmatrix} \sum_{k=1}^3 \epsilon_{1k}a_k & \sum_{k=1}^3 \epsilon_{1k}b_k & \sum_{k=1}^3 \epsilon_{1k}c_k \\ \sum_{k=1}^3 \epsilon_{2k}a_k & \sum_{k=1}^3 \epsilon_{2k}b_k & \sum_{k=1}^3 \epsilon_{2k}c_k \\ \sum_{k=1}^3 \epsilon_{3k}a_k & \sum_{k=1}^3 \epsilon_{3k}b_k & \sum_{k=1}^3 \epsilon_{3k}c_k \\ \end{vmatrix} $$
Evaluating the determinant, we have:
$$ = \sum_{k=1}^3 \epsilon_{1k}a_k [\sum_{k=1}^3 \epsilon_{2k}b_k \sum_{k=1}^3 \epsilon_{3k}c_k - \sum_{k=1}^3 \epsilon_{2k}c_k \sum_{k=1}^3 \epsilon_{3k}b_k] - \sum_{k=1}^3 \epsilon_{1k}b_k [\sum_{k=1}^3 \epsilon_{2k}a_k \sum_{k=1}^3 \epsilon_{3k}c_k - \sum_{k=1}^3 \epsilon_{2k}c_k \sum_{k=1}^3 \epsilon_{3k}a_k] + \sum_{k=1}^3 \epsilon_{1k}c_k [\sum_{k=1}^3 \epsilon_{2k}a_k \sum_{k=1}^3 \epsilon_{3k}b_k - \sum_{k=1}^3 \epsilon_{2k}b_k \sum_{k=1}^3 \epsilon_{3k}a_k]$$
Which, if I am not mistaken, seems to reduce to zero for some reason? I am not sure as to how to proceed from here to prove invariance.