In the specific case of matrices over an algebraically closed field, an element is regular if and only if its Jordan normal form contains a single Jordan block for each eigenvalue. In that case, the centralizer is the set of polynomials of degree less than evaluated at the matrix, and therefore the centralizer has dimension . If the matrix is diagonalisable, then it is regular if and only if there are different eigenvalues. To see this, notice that will commute with any matrix that stabilises each of its eigenspaces. If there are different eigenvalues, then this happens only if is diagonalisable on the same basis as ; in fact is a linear combination of the first powers of, and the centralizer is an algebraic torus of complex dimension ; since this is the smallest possible dimension of a centralizer, the matrix is regular. However if there are equal eigenvalues, then the centralizer is the product of the general linear groups of the eigenspaces of, and has strictly larger dimension, so that is not regular. For a connected compact Lie group, the regular elements form an open dense subset, made up of -conjugacy classes of the elements in a maximal torus which are regular in. The regular elements of are themselves explicitly given as the complement of a set in, a set of codimension-one subtori corresponding to the root system of. Similarly, in the Lie algebra of, the regular elements form an open dense subset which can be described explicitly as adjoint -orbits of regular elements of the Lie algebra of, the elements outside the hyperplanes corresponding to the root system.
Over an infinite field, a regular element can be used to construct a Cartan subalgebra, a self-normalizing nilpotent subalgebra. Over a field of characteristic zero, this approach constructs all the Cartan subalgebras. Given an element, let be the generalized eigenspace of for eigenvalue zero. It is a subalgebra of. Note that is the same as the multiplicity of zero as an eigenvalue of ; i.e., the least integer m such that in the notation in #Definition. Thus, and the equality holds if and only if is a regular element. The statement is then that if is a regular element, then is a Cartan subalgebra. Thus, is the dimension of at least some Cartan subalgebra; in fact, is the minimum dimension of a Cartan subalgebra. More strongly, over a field of characteristic zero,
every Cartan subalgebra of has the same dimension; thus, is the dimension of an arbitrary Cartan subalgebra,
an element x of is regular if and only if is a Cartan subalgebra, and
every Cartan subalgebra is of the form for some regular element.
For a Cartan subalgebra of a complex semisimple Lie algebra with the root system, an element of is regular if and only if it is not in the union of hyperplanes. This is because: for,
For each, the characteristic polynomial of is.
This characterization is sometimes taken as the definition of a regular element.
