invariant forms on representations of compact groups
Let $G$ be a real Lie group. TFAE:

1.
Every real representation of $G$ has an invariant^{} positive definite form, and $G$ has at least one faithful representation^{}.

2.
One faithful representation of $G$ has an invariant positive definite form.

3.
$G$ is compact^{}.
Also, any group satisfying these criteria is reductive, and its Lie algebra^{} is the direct sum^{} of simple algebras and an abelian^{} algebra^{} (such an algebra is often called reductive).
Proof.
$(1)\Rightarrow (2)$: Obvious.
$(2)\Rightarrow (3)$: Let $\mathrm{\Omega}$ be the invariant form on a faithful representation $V$. Let then representation^{} gives an embedding^{} $\rho :G\to \mathrm{SO}(V,\mathrm{\Omega})$, the group of automorphisms^{} of $V$ preserving $\mathrm{\Omega}$. Thus, $G$ is homeomorphic to a closed subgroup of $\mathrm{SO}(V,\mathrm{\Omega})$. Since this group is compact, $G$ must be compact as well.
(Proof that $\mathrm{SO}(V,\mathrm{\Omega})$ is compact: By induction^{} on $dimV$. Let $v\in V$ be an arbitrary vector. Then there is a map, evaluation on $v$, from $\mathrm{SO}(V,\mathrm{\Omega})\to {S}^{dimV1}\subset V$ (this is topologically a sphere, since $(V,\omega )$ is isometric to ${\mathbb{R}}^{dimV}$ with the standard norm). This is a a fiber bundle, and the fiber over any point is a copy of $\mathrm{SO}({v}^{\u27c2},\mathrm{\Omega})$, which is compact by the inductive hypothesis. Any fiber bundle over a compact base with compact fiber has compact total space. Thus $\mathrm{SO}(V,\mathrm{\Omega})$ is compact).
$(3)\Rightarrow (1)$: Let $V$ be an arbitrary representation of $G$. Choose an arbitrary positive definite form $\mathrm{\Omega}$ on $V$. Then define
$$\stackrel{~}{\mathrm{\Omega}}(v,w)={\int}_{G}\mathrm{\Omega}(gv,gw)\mathit{d}g,$$ 
where $dg$ is Haar measure (normalized so that ${\int}_{G}\mathit{d}g=1$). Since $K$ is compact, this gives a well defined form. It is obviously bilinear, b$\mathrm{SO}(V,\mathrm{\Omega})$y the linearity of integration, and positive definite since
$$\stackrel{~}{\mathrm{\Omega}}(gv,gv)={\int}_{G}\mathrm{\Omega}(gv,gv)\mathit{d}g\ge \underset{g\in G}{inf}\mathrm{\Omega}(gv,gv)>0.$$ 
Furthermore, $\stackrel{~}{\mathrm{\Omega}}$ is invariant, since
$$\stackrel{~}{\mathrm{\Omega}}(hv,hw)={\int}_{G}\mathrm{\Omega}(ghv,ghw)\mathit{d}g={\int}_{G}\mathrm{\Omega}(ghv,ghw)d(gh)=\stackrel{~}{\mathrm{\Omega}}(v,w).$$ 
For representation $\rho :T\to \mathrm{GL}(V)$ of the maximal torus $T\subset K$, there exists a representation ${\rho}^{\prime}$ of $K$, with $\rho $ a $T$subrepresentation of ${\rho}^{\prime}$. Also, since every conjugacy class^{} of $K$ intersects any maximal torus, a representation of $K$ is faithful if and only if it restricts to a faithful representation of $T$. Since any torus has a faithful representation, $K$ must have one as well.
Given that these criteria hold, let $V$ be a representation of $G$, $\mathrm{\Omega}$ is positive definite real form, and $W$ a subrepresentation. Now consider
$${W}^{\u27c2}=\{v\in V\mathrm{\Omega}(v,w)=0\forall w\in W\}.$$ 
By the positive definiteness of $\mathrm{\Omega}$, $V=W\oplus {W}^{\u27c2}$. By induction, $V$ is completely reducible.
Applying this to the adjoint representation^{} of $G$ on $\U0001d524$, its Lie algebra, we find that $\U0001d524$ in the direct sum of simple algebras ${\U0001d524}_{1},\mathrm{\dots},{\U0001d524}_{n}$, in the sense that ${\U0001d524}_{i}$ has no proper nontrivial ideals, meaning that ${\U0001d524}_{i}$ is simple in the usual sense or it is abelian. ∎
Title  invariant forms on representations of compact groups 

Canonical name  InvariantFormsOnRepresentationsOfCompactGroups 
Date of creation  20130322 13:23:40 
Last modified on  20130322 13:23:40 
Owner  bwebste (988) 
Last modified by  bwebste (988) 
Numerical id  11 
Author  bwebste (988) 
Entry type  Theorem 
Classification  msc 5400 