Proving Lie Group \rho Preserves Inner Product/Cross Product

[jdstokes]

Let $\rho : \mathbb{H} \to \mathbb{H}; q \mapsto u^{-1}q u$
where u is any unit quaternion. Then $\rho$ is a continuous automorphism of H.

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

The only thing I can think of is that $\rho$ acts on $\mathbf{i}\mathbb{R} + \mathbf{j}\mathbb{R} + \mathbf{k}\mathbb{R}$ 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 $\rho$ rotates $\mathbf{i}\mathbb{R} + \mathbf{j}\mathbb{R} + \mathbf{k}\mathbb{R}$ ?

 

[HallsofIvy]

The obvious method would be to apply the inner product and cross product to $u^{-1}qu$ and show that you get the same thing.

 

[jdstokes]

Hi HallsofIvy,

I tried that but there are 4 x 3 x 4 =48 terms when we come to calculate e.g.

$u^{-1}q u = \bar{u} q u = (u_0 - u_1i-u_2j-u_2k)(p_1 i + p_2j+p_3k)(u_0 + u_1i+u_2j+u_2k)$. Is it really necessary to expand this whole thing out and then take the dot product with another one?