Hyperbolic quaternion

In abstract algebra, the algebra of hyperbolic quaternions is a nonassociative algebra over the real numbers with elements of the form

${\displaystyle q=a+bi+cj+dk,\quad a,b,c,d\in \mathbb {R} \!}$

where the squares of i, j, and k are +1 and distinct elements of {i, j, k} multiply with the anti-commutative property.

The four-dimensional algebra of hyperbolic quaternions incorporates some of the features of the older and larger algebra of biquaternions. They both contain subalgebras isomorphic to the split-complex number plane. Furthermore, just as the quaternion algebra H can be viewed as a union of complex planes, so the hyperbolic quaternion algebra is a pencil of planes of split-complex numbers sharing the same real line.

It was Alexander Macfarlane who promoted this concept in the 1890s as his Algebra of Physics, first through the American Association for the Advancement of Science in 1891, then through his 1894 book of five Papers in Space Analysis, and in a series of lectures at Lehigh University in 1900.