Quaternions

A quaternion is often represented as $q = a + b i + c j + d k$ , where $a, b, c,$ and $d$ are real numbers, and $i, j,$ and $k$ are the fundamental quaternion units. These units have the following multiplication rules:

$i^{2} = j^{2} = k^{2} = i j k = - 1$
$i j = k,$ but $j i = - k$
$j k = i,$ but $k j = - i$
$k i = j,$ but $i k = - j$

Quaternions are often written as $H$ . If we have an unitary quaternion $u$ we have:
- The transformation:
$$
\begin{array}{ccl}
\mathbb{H}& \longrightarrow & \mathbb{H}\
q & \longrightarrow & u \cdot q\
\end

i s a n i s o m e t r y, s i n c e $ | u q - u p | = | u | \cdot | q - p | = | q - p | $ . - T h e t r a n s f o r m a t i o n :

\begin{array}{ccl}
P& \longrightarrow & P\
q & \longrightarrow & u \cdot q\cdot u^{-1}\
\end

i s a r o t a t i o n i n t h e p u r e i m a g i n a r y q u a t e r n i o n s $ P = R i + R j + R k ≅ R^{3} $ - T h e t r a n s f o r m a t i o n :

\begin{array}{ccl}
\mathbb{H}& \longrightarrow & \mathbb{H}\
q & \longrightarrow & -u \cdot \bar{q}\cdot u\
\end

i s a r e f l e c t i o n i n $ H $ a n d v i c e v e r s a . - A n y r o t a t i o n i n $ H = R^{4} $ h a s t h e f o r m :

\begin{array}{ccl}
\mathbb{H}& \longrightarrow & \mathbb{H}\
q & \longrightarrow & v \cdot q\cdot w\
\end

for $v, w$ unitary quaternions. Unitary quaternions are equivalent to the special unitary groupSU(2). Related:quaternions in Geometric Algebra. AsClifford algebras: Observe that $\text{Cl}_{0,2}(\mathbb{R})$ is a four-dimensional algebra spanned by $\{1, e_1, e_2, e_1e_2\}$ which behave like quaternions. Also quaternions can be understood as the even subalgebra of $Cl(3,0)$.