A Lie algebra is a vector space with a Lie bracket , satisfying the Jacobi identity. Hence any element gives a linear transformation given by
(1)

which is called the adjoint representation of . It is a Lie algebra representation because of the Jacobi identity,
(2)
 
(3)
 
(4)

A Lie algebra representation is given by matrices. The simplest Lie algebra is the set of matrices. Consider the adjoint representation of , which has four dimensions and so will be a fourdimensional representation. The matrices
(5)
 
(6)
 
(7)
 
(8)

give a basis for . Using this basis, the adjoint representation is described by the following matrices:
(9)
 
(10)
 
(11)
 
(12)
