View source for Completely-reducible set

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A set  $  M $
of linear operators on a topological vector space  $  E $
with the following property: Any closed subspace in  $  E $
that is invariant with respect to  $  M $
has a complement in  $  E $
that is also invariant with respect to  $  M $. 
In a Hilbert space  $  E $
any set  $  M $
that is symmetric with respect to Hermitian conjugation is completely reducible. In particular, any group of unitary operators is a completely-reducible set. A representation  $  \phi $
of an algebra (group, ring, etc.)  $  A $
is called completely reducible if the set  $  M = \{ {\phi (a) } : {a \in A } \} $
is completely reducible. If  $  A $
is a compact group or a semi-simple connected Lie group (Lie algebra), any representation of  $  A $
in a finite-dimensional vector space is completely reducible (the principle of complete reducibility).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.P. Zhelobenko,   "Compact Lie groups and their representations" , Amer. Math. Soc.  (1973)  (Translated from Russian)</TD></TR></table>

====Comments====
The principle of complete reducibility is commonly referred to as Weyl's theorem (cf. [[#References|[a1]]], Chapt. 2 Sect. 6).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.E. Humphreys,   "Introduction to Lie algebras and representation theory" , Springer  (1972)</TD></TR></table>