Editing Null vector (section)

==Split algebras==
A composition algebra with a null vector is a '''split algebra'''.<ref>Arthur A. Sagle & Ralph E. Walde (1973) ''Introduction to Lie Groups and Lie Algebras'', page 197, [[Academic Press]]</ref>

In a [[composition algebra]] (''A'', +, ×, *), the quadratic form is q(''x'') = ''x x''*. When ''x'' is a null vector then there is no multiplicative inverse for ''x'', and since ''x'' ≠ 0, ''A'' is not a [[division algebra]].

In the [[Cayley–Dickson construction]], the split algebras arise in the series [[bicomplex number]]s, [[biquaternion]]s, and [[bioctonion]]s, which uses the [[complex number]] field <math>\Complex</math> as the foundation of this doubling construction due to [[L. E. Dickson]] (1919). In particular, these algebras have two [[imaginary unit]]s, which commute so their product, when squared, yields +1:
:<math>(hi)^2 = h^2 i^2 = (-1)(-1) = +1 .</math> Then
:<math>(1 + hi)(1 + hi)^* = (1 +hi)(1 - hi) = 1 - (hi)^2 = 0</math> so 1 + hi is a null vector.
The real subalgebras, [[split complex number]]s, [[split quaternion]]s, and [[split-octonion]]s, with their null cones representing the light tracking into and out of 0 ∈ ''A'', suggest [[spacetime topology]].