Continuous linear representation of compact group over reals has invariant symmetric positive-definite bilinear form

For finite-dimensional vector space
Suppose $$G$$ is a compact group and $$\alpha:G \to GL(V)$$ (with $$V$$ finite-dimensional) is a linear representation of $$G$$ over a vector space $$V$$ over $$\R$$, the field of real numbers. Then, there exists a symmetric positive-definite bilinear form $$b: V \times V \to V$$ such that

$$\! b(\alpha(g)v,\alpha(g)w) = b(v,w) \ \forall \ g \in G, \ v,w \in V$$

In other words, we can think of $$b$$ as a dot product invariant under the action of $$G$$.

Further, if we choose a basis for $$V$$ that is an orthonormal basis for $$b$$, then in this basis, all the matrices for $$\alpha(g), g \in G$$, are orthogonal matrices. Thus, another formulation is that every continuous finite-dimensional linear representation over the real numbers is an orthogonalizable linear representation.

Related facts

 * Continuous linear representation of compact group over complex numbers has invariant Hermitian inner product
 * Linear representation of finite group over reals has invariant symmetric positive-definite bilinear form
 * Linear representation of finite group over complex numbers has invariant Hermitian inner product