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The dot product and cross product on \(\mathbb{R}^3\) come from taking the imaginary quaternions \[ \{ (a i + b j + ck) : \; a,b,c \in \mathbb{R} \} \cong \mathbb{R}^3 \] and setting \[ x \cdot y = -\frac{x y + y x}{2} \qquad x \times y = \frac{x y - y x}{2} \]
The same formulas give the imaginary split octonions \(\mathrm{Im}(\mathbb{O}')\) a dot product and cross product.
\(\mathrm{G}_2'\) is the group of linear transformations of \(\mathrm{Im}(\mathbb{O}')\) that preserve both these products!