Null vector

A pseudo-Euclidean vector space may be decomposed (non-uniquely) into orthogonal subspaces A and B, X = A + B, where q is positive-definite on A and negative-definite on B.

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 particular, these algebras have two imaginary units, which commute so their product, when squared, yields +1: The real subalgebras, split complex numbers, split quaternions, and split-octonions, with their null cones representing the light tracking into and out of 0 ∈ A, suggest spacetime topology.

Null vectors are also used in the Newman–Penrose formalism approach to spacetime manifolds.

[3] In the Verma module of a Lie algebra there are null vectors.