the diagonal entries and changing of sign the two other entries. The product of a split-quaternion with its conjugate is the isotropic quadratic form: which is called the norm of the split-quaternion or the determinant of the associated matrix. The real part of a split-quaternion is . It equals the trace of associated matrix. The norm of a product of two split-quaternions is the product of their norms. Equivalently, the determinant of a product of matrices is the product of their determinants. This means that split-quaternions and 2×2 matrices form a composition algebra. As there are nonzero split-quaternions having a zero norm, split-quaternions form a "split composition algebra" – hence their name. A split-quaternion with a nonzero norm has a multiplicative inverse, namely . In terms of matrix, this is Cramer rule that asserts that a matrix is invertible if and only its determinant is nonzero, and, in this case, the inverse of the matrix is the quotient of the cofactor matrix by the determinant. The isomorphism between split-quaternions and 2×2 matrices shows that the multiplicative group of split-quaternions with a nonzero norm is isomorphic with and the group of split quaternions of norm is isomorphic with Representation as complex matrices There is a representation of the split-quaternions as a unital associative subalgebra of the matrices with complex entries. This representation can be defined by the algebra homomorphism that maps a split-quaternion to the matrix Here, (italic) is the imaginary unit, which must not be confused with the basic split quaternion (upright roman). The image of this homomorphism is the matrix ring formed by the matrices of the form where the superscript denotes a complex conjugate. This homomorphism maps respectively the split-quaternions on the matrices The proof that this representation is an algebra homomorphism is straightforward but requires some boring computations, which can be avoided by starting from the expression of split-quaternions as real matrices, and using matrix similarity. Let be the matrix Then, applied to the representation of split-quaternions as real matrices, the above algebra homomorphism is the matrix similarity. It follows almost immediately that for a split quaternion represented as a complex matrix, the conjugate is the matrix of the cofactors, and the norm is the determinant. With the representation of split quaternions as complex matrices. the matrices of quaternions of norm are exactly the elements of the special unitary group SU(1,1). This is used for in hyperbolic