I am reading the book "Group Theoretical Methods and Their Applications" by A. Fassler and E. Stiefel. There is a part that I do not understand.
Suppose $\mathcal{v}$ is a finite representation of group $G$.
Suppose each element of $\mathcal{v}$ (i.e. each linear transformation in $\mathcal{v}$) is described by matrix $D(s)$ for all $s \in G$. Let $D(s) \in GL(V)$ (i.e. the elements of the representation can be described by nonsingular matrices, which are linear transformations from vector space $V$ to itself).
Reducible Representation: If $V$ has proper invariant subspace $V_1$ (i.e. $x \in V_1$ implies $D(s)x \in V_1$ for all $s\in G$), then $\mathcal{v}$ is a reducible representation, and we can write:$$\mathcal{v}=\mathcal{v_1}\oplus\mathcal{v_2}$$ $$V=V_1 \oplus V_2$$
Completely Reducible: If $V$ has proper invariant subspace $V_1$ and $V=V_1 \oplus V_2$, we say $\mathcal{v}$ is completely reducible if $V_2$ is also reducible.
If $\mathcal{v}$ is completely reducible, then there exists a basis so that D(s) for all $s \in G$ become block diagonal (i.e. there exists $T$ such that $T^{-1}D(s)T$ are block diagonal).
Unitary Representation: if $\langle D(s)x,D(s)y \rangle=\langle x,y \rangle$ for all $s \in G$ then $\mathcal{v}$ is a unitary representation.
If $D(s)$ are orthogonal matrices with respect to a basis, namely $\mathcal{B}$, then $\mathcal{v}$ is a unitary representation.
Theorem 1.5 (page 28): Every unitary representation is completely reducible.
Now, suppose $D(s)$ are orthogonal matrices with respect to $\mathcal{B}$, then $\mathcal{v}$ is completely reducible. Since $\mathcal{v}$ is completely reducible, there exists a basis so that D(s) for all $s \in G$ become block diagonal.
Here, I did not understand something: $D(s)$ are orthogonal with respect to $\mathcal{B}$. If we change the basis to make them block diagonal then $D(s)$ can be no more orthogonal and so $\mathcal{v}$ can be no more completely reducible!!!
How can we make $D(s)$ block diagonal when we need to find an appropriate basis, while $D(s)$ are orthogonal in possibly another basis?
The book states that "Every finite-dimensional unitary representation is described by unitary matrices relative to some orthonormal basis."
Should I comprehend the whole story as follows:?
Suppose $D(s)$ are matrices in $\mathcal{v}$. If there exists an orthonormal basis that makes $D(s)$ orthogonal, then there exists a basis that makes them block diagonal.