Editing Hyperbolic quaternion (section)

{{short description|Mutation of quaternions where unit vectors square to +1}}
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In [[abstract algebra]], the [[algebra over a field|algebra]] of '''hyperbolic quaternions''' is a [[nonassociative algebra]] over the [[real numbers]] with elements of the form
:<math>q = a + bi + cj + dk, \quad a,b,c,d \in \mathbb{R} \!</math>
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 [[biquaternion]]s. They both contain subalgebras isomorphic to the [[split-complex number]] plane. Furthermore, just as the quaternion algebra '''H''' can be viewed as a [[quaternion#As a union of complex planes|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.