algebra of biquaternions forms a composition algebra and can be constructed from bicomplex numbers. See below. Place in ring theory Linear representation Note the matrix product . Because h is the imaginary unit, each of these three arrays has a square equal to the negative of the identity matrix. When this matrix product is interpreted as i j = k, then one obtains a subgroup of matrices that is isomorphic to the quaternion group. Consequently, represents biquaternion q = u 1 + v i + w j + x k. Given any 2 × 2 complex matrix, there are complex values u, v, w, and x to put it in this form so that the matrix ring M(2,C) is isomorphic<ref>Leonard Dickson (1914) Linear Algebras, §13 "Equivalence of the complex quaternion and matric algebras"], page 13, via HathiTrust</ref> to the biquaternion ring. Subalgebras Considering the biquaternion algebra over the scalar field of real numbers , the set forms a basis so the algebra has eight real dimensions. The squares of the elements , and are all positive one, for example, . The subalgebra given by is ring isomorphic to the plane of split-complex numbers, which has an algebraic structure built upon the unit hyperbola. The elements and also determine such subalgebras. Furthermore, is a subalgebra isomorphic to the tessarines. A third subalgebra called coquaternions is generated by and . It is seen that , and that the square of this element is . These elements generate the dihedral group of the square. The linear subspace with basis thus is closed under multiplication, and forms the coquaternion algebra. In the context of quantum mechanics and spinor algebra, the biquaternions , and (or their negatives), viewed in the representation, are called Pauli matrices. Algebraic properties The biquaternions have two conjugations: the biconjugate or biscalar minus bivector is and the complex conjugation of biquaternion coefficients where when Note that Clearly, if then is a zero divisor. Otherwise is defined over the complex numbers. Further, is easily verified. This allows an inverse to be defined by , if Relation to Lorentz transformations Consider now the linear subspace is not a subalgebra since it is not closed under products; for example . Indeed, cannot form an algebra if it is not even a magma. Proposition: If is in , then Proof: From the definitions, Definition: Let biquaternion satisfy Then the Lorentz transformation associated with is given