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In mathematics, a regular element of a Lie algebra or Lie group is an element whose centralizer has dimension as small as possible.For example, in a complex semisimple Lie algebra, an element X ∈ g {\displaystyle X\in {\mathfrak {g}}} is regular if its centralizer in g {\displaystyle {\mathfrak {g}}} has dimension equal to the rank of g {\displaystyle {\mathfrak {g}}} , which in turn equals the dimension of some Cartan subalgebra h {\displaystyle {\mathfrak {h}}} .An element g ∈ G {\displaystyle g\in G} a Lie group is regular if its centralizer has dimension equal to the rank of G {\displaystyle G}.