Cartan subalgebras and root decompositions

In this setting a Cartan subalgebra h can be characterized as a maximal toral subalgebra: it is abelian and every ad(H), H in h, is diagonalizable. These commuting actions are simultaneously diagonalizable on g. Their zero joint eigenspace is h itself, and the nonzero joint eigenvalues are linear functionals α on h. The corresponding root space gα consists of vectors X with [H,X]=α(H)X for every H. A root is thus a functional, not the matrix X or a selected arrow in a diagram.

The dimension of h is the rank, and each nonzero root space is one dimensional over ℂ. Cartan subalgebras are conjugate under inner automorphisms, but choosing one is still a coordinate choice in a calculation. A more general Lie algebra definition uses nilpotent self-normalizing subalgebras; its equivalence to the toral description belongs to the present semisimple complex setting. One should not assume that every Cartan subalgebra of an arbitrary solvable or real Lie algebra is a simultaneously diagonalizable complex torus of the same form.

The Jacobi identity shows that a bracket of an α-root vector and a β-root vector has weight α+β. If this sum is nonzero and not a root, that bracket vanishes. If β=−α, the bracket lies in h and is nonzero for suitable nonzero root vectors. They can be scaled to produce eα,fα,hα with [eα,fα]=hα and α(hα)=2, forming an sl(2) triple. The element hα is the coroot in h, while α remains an element of h*.

A positive-root system, introduced by choosing a chamber, divides roots into positive and negative ones. It determines n+ and n− and hence a triangular decomposition. This is additional structure beyond h. It will select which vectors are called highest and which root operators raise weights. Reversing the positivity convention exchanges those roles without changing the underlying algebra. Record these choices before constructing a highest-weight module or comparing formulas from two sources, especially when their upper- versus lower-triangular conventions differ.

An element H in h is regular when α(H) is nonzero for every root. The root decomposition then shows that its centralizer in g is exactly h. If H lies on a root hyperplane, extra root spaces centralize it and the centralizer grows. Thus belonging to a Cartan subalgebra does not automatically make an element regular. This distinction is visible even in diagonal matrix examples with repeated diagonal entries, where blocks of off-diagonal matrices begin to commute with the selected element.

Root decomposition organizes an algebraic representation, not a Euclidean spectral theorem for one real matrix. To draw roots as arrows one later chooses a real span and an invariant positive inner product on that span. The complex Killing form itself is not a positive Hermitian metric. Keeping h, its dual, root vectors, and coroots distinct makes those geometric drawings legitimate coordinate realizations rather than ambiguous identifications. The payoff is that large bracket tables become constrained by a small set of root additions and rank-one relations.