Let [itex]\rho : \mathbb{H} \to \mathbb{H}; q \mapsto u^{-1}q u[/itex](adsbygoogle = window.adsbygoogle || []).push({});

where u is any unit quaternion. Then [itex]\rho[/itex] is a continuous automorphism of H.

I'm asked to show that [itex]\rho[/itex] preserves the inner product and cross product on the subspace [itex]\mathbf{i}\mathbb{R} + \mathbf{j}\mathbb{R} + \mathbf{k}\mathbb{R}[/itex] consisting of purely imaginary quaternions.

The only thing I can think of is that [itex]\rho[/itex] acts on [itex]\mathbf{i}\mathbb{R} + \mathbf{j}\mathbb{R} + \mathbf{k}\mathbb{R}[/itex] by rotating that subspace (for which I know a proof), and rotations preserve angles and orientation.

Is there a more direct method which avoids using the fact that [itex]\rho[/itex] rotates [itex]\mathbf{i}\mathbb{R} + \mathbf{j}\mathbb{R} + \mathbf{k}\mathbb{R}[/itex] ?

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# Homework Help: Lie groups question

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