Lie Groups and Lie Algebras: Winter-2023 Practice Exercises 8 : Due Friday 3/10
Octonionic rotations
S0 4617S
Choose two elements \(x,y\in\mathbb{O}\) such that \(xy\ne yx\) and
\(x,y\not\in\mathbb{H}\subset\mathbb{O}\).
(The notation is such that \(\mathbb{H}=\langle 1,i,j,k \rangle\) and
\(\mathbb{O}=\mathbb{H}\oplus\mathbb{H}\ell\).)
Compute the following quantities:
\begin{align*}
a &= \bar{\imath}\bigl(\bar{\jmath}(\bar{k}x)\bigr) \\
b &= \bigl((xk)j\bigr)i \\
c &= \bar\imath\bigl(\bar\jmath(\bar{k}xk)j\bigr)i \\
p &= \bar{\imath}\bigl(\bar{\jmath}(\bar{k}y)\bigr) \\
q &= \bigl((yk)j\bigr)i \\
r &= \bar\imath\bigl(\bar\jmath(\bar{k}yk)j\bigr)i
\end{align*}
What do you notice about your results?
Compute the following quantities:
\begin{align*}
s &= xy \\
t &= \bigl((sk)j\bigr)i \\
u &= ap
\end{align*}
What do you notice about your results?