Consider the operator $S = \sum_{i,j} X_{ij}^2$ on $L^2(SO(n+1))$, where $X_{ij}$ generates the rotation of the sphere $S^n$ in the $ij$-plane keeping the $(n - 2)$ complementary directions fixed. Consider the operator $-S + X^2 - i\lambda X$, where $X$ is any one of the $X_{ij}$'s. I want to see when this operator is positive semi-definite (meaning, for what range of $\lambda$). I think there is a subelliptic estimate for the operator $-S + X^2$, but it is the $\lambda X$ part that is bothering me.

It might be possible that an answer is only possible for certain values of $n$. Any hints would be well-appreciated. Also, could someone point out a source where I can read about such problems?

Thanks in advance!!