Maximal torus ($T$)

A maximal torus is a connected compact abelian Lie subgroup maximal among such subgroups. Restricting a finite-dimensional unitary representation to it splits into one-dimensional weight characters. For special unitary group $\mathrm{SU}(2)$ it is the diagonal circle $\mathrm{diag}(z,z^{-1})$.

Give the Lie algebra of a compact Lie group an inner product invariant under the Adjoint representation. For a maximal torus $T$ with Lie algebra $\mathfrak{t}$, its Lie-algebra centralizer is $\mathfrak{t}$: any additional commuting vector would generate, together with $T$, a larger compact connected abelian subgroup. The torus weight decomposition therefore supplies $H\in \mathfrak{t}$ whose centralizer is exactly $\mathfrak{t}$, by avoiding finitely many nonzero-weight hyperplanes. For any $X$, maximize $⟨{\mathrm{Ad}}_{g}X,H⟩$ on the group. Its directional derivative gives $⟨Z,[{\mathrm{Ad}}_{g}X,H]⟩=0$ for every $Z$, so ${\mathrm{Ad}}_{g}X\in \mathfrak{t}$. For a connected compact group, the bi-invariant Riemannian metric and Hopf-Rinow theorem make the exponential onto; exponentiating this conjugacy puts every element in a conjugate of $T$.

For a compact torus $T\cong U(1{)}^r$, its continuous one-dimensional characters form a free abelian group of rank $r$. They are precisely $z\mapsto z_1^{m_1}\cdots z_r^{m_r}$ with integer exponents. This follows from the classification of representations of the circle group, applied to each factor. These characters are the integral group weights occurring upon restricting a group representation to a maximal torus; global integrality must be distinguished from integrality against coroots for a simply connected semisimple covering group.