Seven-dimensional Cross Product - Rotations

Rotations

In three dimensions the cross product is invariant under the group of the rotation group, SO(3), so the cross product of x and y after they are rotated is the image of x × y under the rotation. But this invariance is not true in seven dimensions; that is, the cross product is not invariant under the group of rotations in seven dimensions, SO(7). Instead it is invariant under the exceptional Lie group G₂, a subgroup of SO(7).