Split-octonion

In mathematics, the split-octonions are an 8-dimensional nonassociative algebra over the real numbers. Unlike the standard octonions, they contain non-zero elements which are non-invertible. Also the signatures of their quadratic forms differ: the split-octonions have a split signature (4,4) whereas the octonions have a positive-definite signature (8,0). Up to isomorphism, the octonions and the split-octonions are the only two 8-dimensional composition algebras over the real numbers. They are also the only two octonion algebras over the real numbers. Split-octonion algebras analogous to the split-octonions can be defined over any field. The octonions and the split-octonions can be obtained from the Cayley–Dickson construction by defining a multiplication on pairs of quaternions. We introduce a new imaginary unit l and write a pair of quaternions (a, b) in the form a + lb. The product is defined by the rule: where If λ is chosen to be −1, we get the octonions. If, instead, it is taken to be +1 we get the split-octonions. One can also obtain the split-octonions via a Cayley–Dickson doubling of the split-quaternions. Here either choice of λ (±1) gives the split-octonions. A basis for the split-octonions is given by the set . Every split-octonion can be written as a linear combination of the basis elements, with real coefficients . By linearity, multiplication of split-octonions is completely determined by the following multiplication table: A convenient mnemonic is given by the diagram at the right, which represents the multiplication table for the split-octonions. This one is derived from its parent octonion (one of 480 possible), which is defined by: where is the Kronecker delta and is the Levi-Civita symbol with value when and: with the scalar element, and The red arrows indicate possible direction reversals imposed by negating the lower right quadrant of the parent creating a split octonion with this multiplication table. The conjugate of a split-octonion x is given by just as for the octonions.