Fields generated by sums and products of singular moduli

We show that the field $\mathbb{Q}(x,y)$, generated by two singular moduli~$x$ and~$y$, is generated by their sum $x+y$, unless~$x$ and~$y$ are conjugate over~$\mathbb{Q}$, in which case $x+y$ generates a subfield of degree at most~$2$. We obtain a similar result for the product of two singular moduli.