<s>[INST] What is $\left( \frac{t^2}{t^2+2}i+\frac{2}{t^2+2}j+\frac{2t}{t^2+2}k \right) \times \frac{2ti-2tj-(t^2+2)k}{t^2+2}$? I did
\begin{align}
& \left( \frac{t^2}{t^2+2}i+\frac{2}{t^2+2}j+\frac{2t}{t^2+2}k \right) \times \frac{2ti-2tj-(t^2+2)k}{t^2+2} \\[8pt]
= {} & \frac{2t^2i-4tj-2t(t^2+2)k}{(t^2+2)^2} \\[8pt]
= {} & \frac{2t^2i-4tj-2t^3k+4tk}{(t^2+2)^2} \\[8pt]
= {} & \frac{-4tj+4tk}{(t^2+2)^2}
\end{align}
Is this true?
 [/INST] \begin{align}
i\times j & = k = -j\times i \\
j\times k & = i = -k\times j \\
k\times i & = j = - i\times k \\
i \times i & = j \times j = k\times k = 0
\end{align}
</s>