The operator for linear momentum $\mathbf{p}$ and the operator for orbital angular momentum $\mathbf{L}$ ($\mathbf{L} = \mathbf{r} \times \mathbf{p}$) are Hermitian. Is the cross product between $\mathbf{p}$ and $\mathbf{L}$ also Hermitian? If I go component wise and use the commutator bracket relation $[p_i,L_j]=i\hbar \epsilon _{ijm}p_m$, then I can show that the cross product is not Hermitian. But on using the vector triple product relation, I see that it's Hermitian indeed. Can someone explain to me where am I going wrong?
Also, can we state this in general for any such cross product between two Hermitian operators?