Simpler proof of the lemma: if $\gcd(a,b)=1$ then all odd factor of $a^2 + 3b^2$ has the same form?

Question

Lemma: If $\gcd(a,b)=1$, then every odd factor of $a^2 + 3b^2$ has this same form.

I was reviewing the proofs for this lemma online. Every proof is long and cumbersome. Is there a simpler method to prove it?