View source for Lie group, derived

The [[Commutator subgroup|commutator subgroup]] of a [[Lie group|Lie group]]. For any Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058620/l0586201.png" /> its derived Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058620/l0586202.png" /> is a normal (not necessarily closed) Lie subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058620/l0586203.png" />. The corresponding ideal of the [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058620/l0586204.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058620/l0586205.png" /> coincides with the commutator algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058620/l0586206.png" /> (also called the derived Lie algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058620/l0586207.png" />). The commutator subgroup of a simply-connected (or connected linear) Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058620/l0586208.png" /> is always closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058620/l0586209.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Chevalley,   "Theory of Lie groups" , '''1''' , Princeton Univ. Press  (1946)</TD></TR></table>