Editing Maximal torus (section)

==Examples==
The [[unitary group]] U(''n'') has as a maximal torus the subgroup of all [[diagonal matrices]]. That is,
: <math>T = \left\{\operatorname{diag}\left(e^{i\theta_1},e^{i\theta_2},\dots,e^{i\theta_n}\right) : \forall j, \theta_j \in \mathbb{R}\right\}.</math>
''T'' is clearly isomorphic to the product of ''n'' circles, so the unitary group U(''n'') has rank ''n''. A maximal torus in the [[special unitary group]] SU(''n'') ⊂ U(''n'') is just the intersection of ''T'' and SU(''n'') which is a torus of dimension&nbsp;''n''&nbsp;&minus;&nbsp;1.

A maximal torus in the [[special orthogonal group]] SO(2''n'') is given by the set of all simultaneous [[rotation]]s in any fixed choice of ''n'' pairwise orthogonal planes (i.e., two dimensional vector spaces). Concretely, one maximal torus consists of all block-diagonal matrices with <math>2\times 2</math> diagonal blocks, where each diagonal block is a rotation matrix.
This is also a maximal torus in the group SO(2''n''+1) where the action fixes the remaining direction. Thus both SO(2''n'') and SO(2''n''+1) have rank ''n''. For example, in the [[rotation group SO(3)]] the maximal tori are given by rotations about a fixed axis.

The [[symplectic group]] Sp(''n'') has rank ''n''. A maximal torus is given by the set of all diagonal matrices whose entries all lie in a fixed complex subalgebra of [[Quaternion|'''H''']].