Lie群

 group is a Lie gp. when
(i) smooth mfd.
(ii)  smooth

 m-dim abelian Lie gp.
 compact m-dim abelian Lie gp.

 n^2-dim Lie gp.
  as  n^2-dim Lie ring
 n(n-1)/2-dim Lie gp.
  as  n(n-1)/2-dim Lie ring
 (n^2-1)-dim Lie gp.
  as  (n^2-1)-dim Lie ring