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It's not clear why the quaternions are closed under addition. All of the constructions I've seen make it clear why they're closed under multiplication, but not addition.

Anyway, consider the following sandwich product $$\begin{aligned} (q + r)^*v(q + r) &= q^*vq + q^*vr + r^*vq + r^*vr\\ &= q^*vq + q^*vr + (q^*v^*r)^* + r^*vr\\ &= q^*vq + q^*vr - (q^*vr)^* + r^*vr\\ &= q^*vq + 2\operatorname{Vec}(q^*vr) + r^*vr \end{aligned}$$

It's evident that this is a vector. But nevertheless, what does $$\operatorname{Vec}(q^*vr)$$ mean? Does it show up anywhere else?

It seems natural when you express it as $$\frac12[(q+r)^*v(q+r) - q^*vq - r^*vr] $$ which looks similar to $$a\cdot b = \frac12(|a+b|^2 - |a|^2 - |b|^2)$$

It's clear that it's linear in $v$, which narrows down what it could be.

Let's call it $L(v; q, r)$.

We have that $L(Q^*vQ;q,r) = L(v; Qq, Qr)$

Different values of $p$ and $q$ span different families of transformations on $v$. Examples include rotation, roto-reflection, cross product...

$L$ is tri-linear.

It looks similar to Singular Value Decomposition.

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