8. Iterated commutators and derivatives of the Levi form

Problem 8.1.
[J. Dâ€™Angelo] Let M be a pseudoconvex CR manifold of hypersurface type. Let L be a (1,0) vector field. Let $\lambda(L,\bar L)$ denote the Levi form on $L$. Consider two notions: the type of $L$ at $p$ (via iterated commutators getting into the bad direction) and the number $c(L,p)$ defined by 2 + the order of vanishing of $\lambda(L,\bar L)$ in the directions of $L$ and $\bar L$. Are these the same for all $L$?
It should be noticed that, when the Levi form has eigenvalues of opposite signs at $p$, there is always an $L$ for which these numbers are different.
Cite this as: AimPL: Analysis and geometry on pseudohermitian manifolds, available at http://aimpl.org/pshermitian.