In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers. After introduction in the 20th century of coordinate-free definitions of rings and algebras, it was proved that the algebra of split-quaternions is isomorphic to the ring of the real matrices. So the study of split-quaternions can be reduced to the study of real matrices, and this may explain why there are few mentions of split-quaternions in the mathematical literature of the 20th and 21st centuries. Definition The split-quaternions are the linear combinations (with real coefficients) of four basis elements that satisfy the following product rules: , , , . By associativity, these relations imply , , and also . So, the split-quaternions form a real vector space of dimension four with as a basis. They form also a noncommutative ring, by extending the above product rules by distributivity to all split-quaternions. Let consider the square matrices They satisfy the same multiplication table as the corresponding split-quaternions. As these matrices form a basis of the two by two matrices, the function that maps to (respectively) induces an algebra isomorphism from the split-quaternions to the two by two real matrices. The above multiplication rules imply that the eight elements form a group under this multiplication, which is isomorphic to the dihedral group D4, the symmetry group of a square. In fact, if one considers a square whose vertices are the points whose coordinates are or , the matrix is the clockwise rotation of the quarter of a turn, is the symmetry around the first diagonal, and is the symmetry around the axis. Properties Like the quaternions introduced by Hamilton in 1843, they form a four dimensional real associative algebra. But like the matrices and unlike the quaternions, the split-quaternions contain nontrivial zero divisors, nilpotent elements, and idempotents. (For example, is an idempotent zero-divisor, and is nilpotent.) As an algebra over the real numbers, the algebra of split-quaternions is isomorphic to the algebra of 2×2 real matrices by the above defined isomorphism. This isomorphism allows identifying each split-quaternion with a 2×2 matrix. So every property of split-quaternions corresponds to a similar property of matrices, which is often named differently. The conjugate of a split-quaternion , is . In term of matrices, the conjugate is the cofactor matrix obtained by exchanging