In abstract algebra, the algebra of hyperbolic quaternions is a nonassociative algebra over the real numbers with elements of the form 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 union of split-complex number planes 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. Algebraic structure Like the quaternions, the set of hyperbolic quaternions form a vector space over the real numbers of dimension 4. A linear combination is a hyperbolic quaternion when and are real numbers and the basis set has these products: 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, , while . 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, the set forms a quasigroup. One also notes that any subplane of the set M of hyperbolic quaternions that contains the real axis forms a plane of split-complex numbers. If is the conjugate of , then the product is the quadratic form used in spacetime theory. In fact, for events p and q, the bilinear form arises as the negative of the real part of the hyperbolic quaternion product pq*, and is used in Minkowski space. Note that the set of units U = {q : qq* ≠ 0 } is not closed under multiplication. See the references (external link) for details. Discussion The hyperbolic quaternions form a nonassociative ring; the failure of associativity in this algebra curtails the facility of this algebra in transformation theory. Nevertheless, this algebra put a focus on analytical kinematics by suggesting