A Cartan subgroup of a reductive Lie group 
is the full centralizer in
of a Cartan
subalgebra 
of its Lie algebra. Explicitly, it consists of all
such that 
for every 
, where 
is the adjoint representation.
It need not be connected (Knapp 2002, p. 487).
For a connected complex reductive
algebraic group, Cartan subgroups are maximal tori in the algebraic-group sense,
isomorphic to 
, and are conjugate.
Here 
denotes the nonzero complex numbers under multiplication.
Cartan Subgroup
See also
Lie Group, Maximal Torus, Nilpotent GroupExplore with Wolfram|Alpha
References
Knapp, A. W. Lie Groups Beyond an Introduction, 2nd ed. Boston, MA: Birkhäuser, 2002.Cite this as:
Weisstein, Eric W. "Cartan Subgroup." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CartanSubgroup.html