Quaternion Rotation and Product

Question

I am reading a dissertation where there's a part about quaternion rotation:

Let $w = r_1 e^{\theta _1 i} + r_2 e^{\theta _2 i} j \in \mathbb{H}$ and $q = e^{i \alpha}$

So, the book concludes that:

$r_q \left( w \right) = q w q = r_1 e^{\left(\theta _1 + 2 \alpha \right) i} + r_2 e^{\left(\theta _2 + 2 \alpha \right) i} j$

Wich is a rotation of $w$ by $2\alpha$ in the plane $1i$. But when i try to solve this product, i found:

$r_q \left( w \right) = q w q = r_1 e^{\left(\theta _1 + 2 \alpha \right) i} + r_2 e^{\theta _2 i} j$

Answer 1

Let's correct the source's errors. From the bottom of page 7:

Now, let $w=r_1e^{i\theta_1}+r_2e^{i\theta_2}j\in\mathbb{K}$ and $q=e^{i\alpha}$. Then $$ \mathbf{r}_q(w):= qwq=r_1e^{i\theta_1+2\alpha}+r_2e^{i(\theta_2\color{Red}{+2\alpha})}j, $$ which corresponds to a rotation of $w$ by $2\alpha$ in the plane $1i$. Similarly, $e^{j\alpha}$ and $e^{k\alpha}$ yields rotations in the plane $1k$ and $1j$. Since $\mathrm{SO}(4)$ is generated by such rotations, [...]

The equation should instead read

$$ \mathbf{r}_q(w):=qwq=r_1e^{i\theta_1+2\alpha}+r_2e^{i\theta_2}j $$

Next, on the bottom of page 10:

Let $ x=r_1e^{i\theta_1}+r_2e^{i\theta_2}j+s_1e^{i\phi_1}l+s_2(e^{i\phi_2}j)l\in\mathbb{O}.$

Then $ e^{l\alpha}xe^{l\alpha}=r_1e^{i(\theta_1+2\alpha)}+r_2e^{i\theta_2}j+s_1e^{i(\phi_1+2\alpha)}l+s_2(e^{i\theta_2}j)l,$

which corresponds to a rotation by $2\alpha$ in the plane $1l$. Similarly for rotations in the planes $1i$, $1j$, and $1k$. Hence [...]

For some reason, they decomposed $x$ into planes based on $i$, but used $e^{l\alpha}$ for their rotation. There are two ways to fix this: use $e^{i\alpha}$ instead of $e^{l\alpha}$, or else decompose $x$ into planes based on $l$.

The first way to fix it would look like this:

Let $ x=r_1e^{i\theta_1}+r_2e^{i\theta_2}j+s_1e^{i\phi_1}l+s_2(e^{i\phi_2}j)l\in\mathbb{O}.$

Then $ e^{i\alpha}xe^{i\alpha}=r_1e^{i(\theta_1+2\alpha)}+r_2e^{i\theta_2}j+s_1e^{i\phi_1}l+s_2(e^{i\theta_2}j)l,$

The second way to fix it would look like this:

Let $x=r_1e^{l\theta_1}+r_2e^{l\theta_2}i+r_3e^{l\theta_3}j+r_4e^{l\theta_4}k.$

Then $e^{l\alpha}xe^{l\alpha}=r_1e^{l(\theta_1+2\alpha)}+r_2e^{l\theta_2}i+r_3e^{l\theta_3}j+r_4e^{l\theta_4}k$.

In the second case, we're no longer using the privileged copy of $\mathbb{H}\subset\mathbb{O}$ so I see no reason to split angles and coefficients into two pairs. (Arguably, there wasn't much reason to in the first place.)

Note, depending on how you define/introduce the octonions, you are not stuck with just plane rotations based in $i,j,l,$ etc. $-$ using properties of octonions, we can show $ab=ba$ iff $a,b$ have parallel imaginary parts, and $ab=-ba$ iff they are perpendicular and pure imaginary; then we can show the above formulas for any choice of planes containing the real axis without having to decompose into an unnecessary number of parts with respect to a fixed coordinate system. (It is also very interesting to characterize when $(ab)c=-a(bc)$ in $\mathbb{O}$.)