Tori and maximal tori

Let $G$ be a compact Lie group with identity $e$ (Lie group).

• A torus is a compact connected abelian Lie group. The basic example is the circle group $S^1=\mathbb{R}\mathrm{/}\mathbb{Z}$; the product of $r$ copies of $S^1$ is a torus of dimension $r$.
• A subgroup of $G$ is a subgroup in the algebraic sense; by a Lie subgroup we mean a subgroup that is an immersed, embedded or closed Lie subgroup in the sense of Immersed, embedded, and closed Lie subgroups. A torus of $G$ is a subgroup $T\le G$ that is a torus in the above sense and is a closed (equivalently, embedded) Lie subgroup of $G$.
• A maximal torus of $G$ is a torus $T\le G$ that is maximal under inclusion among torus subgroups of $G$: no torus subgroup of $G$ properly contains $T$. Maximality is with respect to inclusion of subgroups, not with respect to dimension or Lie algebra alone.

Since $G$ is a finite-dimensional real Lie group, a closed subgroup of $G$ is an embedded Lie subgroup (this is Cartan's closed subgroup theorem, used later on this page); in particular every maximal torus is a compact connected abelian embedded Lie subgroup, and its Lie algebra is a subspace of $\mathrm{Lie}G$.

• A torus is a compact connected abelian Lie group; the compact abelian topological group ${\mathbb{Z}}_p$ for a prime $p$ is not a torus because it is totally disconnected (and is not a positive-dimensional Lie group), and a disconnected compact abelian Lie group such as $\mathbb{Z}\mathrm{/}2$ is not a torus either.
• Connectedness is part of the definition: the orthogonal group $O(2)$ is a compact Lie group that is not connected, and its identity component $SO(2)$ is its unique maximal torus; the reflection component contains no torus.
• In an abelian compact Lie group every subgroup is normal. If $A$ is compact abelian and possibly disconnected, every torus in $A$ is connected and hence lies in the identity component $A^0$, which is itself a torus; so $A^0$ is the unique maximal torus of $A$ in that case. Maximality here always means largest in the inclusion order among torus subgroups.