These commutation relations, and the vector space of generators, fulfill the definition of the Lie algebra s o ( 3 , 1 ) . In summary, a Lie algebra is defined as a vector space V over a field of numbers, and with a binary operation [, ] (called a Lie bracket in this context) on the elements of the vector space, satisfying the axioms of bilinearity, alternatization, and the Jacobi identity. Here the operation [, ] is the commutator which satisfies all of these axioms, the vector space is the set of Lorentz generators V as given previously, and the field is the set of real numbers.