Is the Ricci curvature of the compact symplectic group $Sp(n)$ bounded below by $cn$ for some constant $c > 0$ independent of $n$?

For $O(n)$ and $U(n)$ I know many references which state such a bound on Ricci curvature, although none of them actually include complete proofs. (Pointers to such proofs for $O(n)$ and $U(n)$ would also be appreciated.) The usual practice for these groups is merely to refer to Cheeger and Ebin's book, which develops enough general theory of curvature of Lie groups that carrying out the calculations is presumably a straightforward exercise, for those who are on top of such things. (As far as I can see, Cheeger and Ebin don't even state the results in these particular cases.) It's been about ten years since I've been up on such things, which is why I'm hoping someone here knows the answer instead of just trying to work it out myself.