Geometry of quaternionic and pseudo-quaternionic multiplications (Q1062109)

The real pseudo-quaternionic algebra is a 4-dimensional, real, non- associative and non-commutative algebra with multiplication closely related to that of the real quaternions. The present paper deals only briefly with pseudo-quaternions. It is mainly concerned with an investigation of the geometric properties of the linear transformations on a 4-dimensional real vector space, V, induced by the right and left quaternion multiplications on V. Let these multiplications be denoted by \(r_ q\) and \(l_ q\) respectively \((r_ q(x)=x\cdot q\), \(l_ q(x)=q\cdot x\), where \(\cdot\) is the quaternion multiplication) and let \(q'=p^{-1}\cdot q^{-1}\cdot p\), where q is any non-real quaternion and p any non-zero quaternion. A typical result is the following: The linear transformation \(l_ qr_{q'}\) \((l_ qr_{q'}(x)=q\cdot x\cdot q')\) represents a rotation in V about the quaternion p.