Let $ K$ be an imaginary quadratic field and $ E$ be an elliptic curve with CM by $ \mathcal{O}_K$ . Is there a way to see that $ K(j(E), h(E[\mathfrak{p}]))/K$ is an Abelian extension for some $ \mathfrak{p}$ where $ h$ is the Weber function and $ j(E)$ is the $ j$ -invariant, without invoking the fact that it is the ray class field?

I am looking for a proof which is more elementary and probably along the lines of showing that $ Gal(K(j(E), h(E[\mathfrak{p}]))/K)$ injects into an Abelian group, sort of like how we prove $ K(j(E), E_{tors})/ K(j(E))$ is abelian.

Any proof I have found in literature involves showing that it is the ray class field of $ K$ modulo $ \mathfrak{p}$ using a characterization involving the splitting of primes.