Editing Split-quaternion (section)

== Definition ==

The ''split-quaternions'' are the [[linear combination]]s (with real coefficients) of four basis elements {{math|1, i, j, k}} that satisfy the following product rules:
: {{math|1=i<sup>2</sup> = −1}}, 
: {{math|1=j<sup>2</sup> = 1}}, 
: {{math|1=k<sup>2</sup> = 1}},
: {{math|1=ij = k = −ji}}.
By [[associative property|associativity]], these relations imply
: {{math|1=jk = −i = −kj}}, 
: {{math|1=ki = j = −ik}},
and also {{math|1=ijk = 1}}.

So, the split-quaternions form a [[real vector space]] of dimension four with {{math|{{mset|1, i, j, k}}}} as a [[basis (linear algebra)|basis]]. They form also a [[noncommutative ring]], by extending the above product rules by [[distributivity]] to all split-quaternions.

The square matrices
: <math>\begin{align}
\boldsymbol{1} =\begin{pmatrix}1&0\\0&1\end{pmatrix},\qquad&\boldsymbol{i} =\begin{pmatrix}0&1\\-1&0\end{pmatrix},\\
\boldsymbol{j} =\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad&\boldsymbol{k} =\begin{pmatrix}1&0\\0&-1\end{pmatrix}.
\end{align}</math>
satisfy the same multiplication table as the corresponding split-quaternions. As these matrices form a basis of the two-by-two matrices, the unique linear [[function (mathematics)|function]] that maps {{math|1, i, j, k}} to <math>\boldsymbol{1}, \boldsymbol{i}, \boldsymbol{j}, \boldsymbol{k}</math> (respectively) induces an [[algebra isomorphism]] from the split-quaternions to the two-by-two real matrices. 

The above multiplication rules imply that the eight elements {{math|1, i, j, k, −1, −i, −j, −k}} form a [[group (mathematics)|group]] under this multiplication, which is [[isomorphic]] to the [[dihedral group]] D<sub>4</sub>, the [[Examples of groups#The symmetry group of a square: dihedral group of order 8|symmetry group of a square]]. In fact, if one considers a square whose vertices are the points whose coordinates are {{math|0}} or {{math|1}}, the matrix <math>\boldsymbol{i}</math> is the clockwise rotation of the quarter of a turn, <math>\boldsymbol{j}</math> is the symmetry around the first diagonal, and <math>\boldsymbol{k}</math> is the symmetry around the {{mvar|x}} axis.