Editing Hyperbolic quaternion (section)

==Algebraic structure==
Like the [[quaternions]], the set of hyperbolic quaternions form a [[vector space]] over the [[real numbers]] of [[dimension]] 4.  A [[linear combination]]

:<math>q = a+bi+cj+dk</math>

is a '''hyperbolic quaternion''' when  <math>a, b, c,</math> and <math>d</math> are real numbers and the basis set <math>\{1,i,j,k\}</math> has these products:

:<math>ij=k=-ji</math>
:<math>jk=i=-kj</math>
:<math>ki=j=-ik</math>
:<math>i^2=j^2=k^2=+1</math>

Using the [[distributive property]], these relations can be used to multiply any two hyperbolic quaternions.

Unlike the ordinary quaternions, the hyperbolic quaternions are not [[associative]]. For example, <math>(ij)j = kj = -i</math>, while <math>i(jj) = i</math>. In fact, this example shows that the hyperbolic quaternions are not even an [[alternative algebra]].

The first three relations show that products of the (non-real) basis elements are [[anti-commutative]].  Although this basis set does not form a [[group (mathematics)|group]], the set

:<math>\{1,i,j,k,-1,-i,-j,-k\}</math>

forms a [[Quasigroup#Loops|loop]], that is, a [[quasigroup]] with an identity element. One also notes that any subplane of the set ''M'' of hyperbolic quaternions that contains the real axis forms a plane of [[split-complex number]]s. If

:<math>q^*=a-bi-cj-dk</math>

is the conjugate of <math>q</math>, then the product

:<math>q(q^*)=a^2-b^2-c^2-d^2</math>

is the [[quadratic form]] used in [[spacetime]] theory. In fact, for events ''p'' and ''q'', the [[bilinear form]] 
: <math>\eta (p,q) = -p_0q_0 + p_1q_1 + p_2q_2 + p_3q_3 </math>
arises as the negative of the real part of the hyperbolic quaternion product ''pq''*, and is used in [[Minkowski space#Minkowski metric|Minkowski space]].

Note that the set of [[unit (ring theory)|units]] U = {''q'' : ''qq''* ≠ 0 } is ''not'' closed under multiplication. See the references (external link) for details.