Journal of the
Korean Mathematical Society JKMS

We review the algebraic structure of $\mathbb{H}^{^{\prime }}$ and show that $\mathbb{H}^{^{\prime }}$ has a scalar product that allows as to identify it with semi Euclidean $\mathbb{E}_{2}^{4}$. We show that a pair $q$ and $p$ of unit split quaternions in $\mathbb{H}^{^{\prime }}$ determines a rotation $% R_{qp}:\mathbb{H}^{^{\prime }}\rightarrow \mathbb{H}^{^{\prime }}$. Moreover, we prove that $R_{qp}$ is a product of rotations in a pair of orthogonal planes in $\mathbb{E}_{2}^{4}$. To do that we call upon one tool from the theory of second ordinary differential equations.

Keywords: hyperbolic number, split quaternion, generalized inverse, rotation, timelike plane of index 1, timelike plane of index 2, spacelike plane

MSC numbers: Primary 15A33, 15A66