*Remark: I [cross-posted](https://math.stackexchange.com/questions/4277823/existence-of-orthonormal-basis-for-l2g-in-c-cg) this question on MSE and added a bounty to it.*

Suppose that $G$ is a locally compact (Hausdorff) group. It is well-known that the compactly supported functions $C_c(G)$ are dense in $L^2(G)$. In the book 

>"*Operator algebras, theory of $C^{*}$-algebras and von Neumann algebras*" 

written by Bruce Blackadar it is claimed (without proof) that $L^2(G)$ admits an orthonormal basis contained in $C_c(G)$.

I didn't immediately see why this is true. So I started to look for an argument and encountered an [MSE-post](https://math.stackexchange.com/questions/201119/a-complete-orthonormal-system-contained-in-a-dense-sub-space), which shows that it is not always possible to find an orthonormal basis for a non-separable Hilbert space in a given dense subspace. 

I know that Blackadar's claim is true in the following cases:

 - If $G$ is second countable, then $L^{2}(G)$ is separable. So one can use the Gram-Schmidt procedure to find an orthonormal basis in $C_{c}(G)$.
 - If $G$ is compact and abelian, then $\widehat{G}$ (= Pontryagin dual) is an orthonormal basis for $L^{2}(G)$. Note that $\widehat{G}$ can be viewed as a subset of $C_{c}(G)$ in this case.

But does anyone know why Blackadar's claim is true (or false) for general $G$? Or does anyone know a reference for this?