Let us assume the principal symbol of a nonlinear differential operator $E$ at a point $p$ is $$\sigma_{E}(p):\Gamma (T^*M)\times \Gamma (T^*M)\to \Gamma (T^*M)$$ which acts as follows: \begin{align} \sigma_{E}(p) (a,\eta) &= \sum_{k,l=1}^n \Big( a_k a_l\alpha_i\eta_j + a_k a_l \eta_i\alpha_j -a_i a_k\alpha_l\eta_j- a_i a_k\eta_l\alpha_j\\ &-a_ja_k\alpha_l \eta_i-a_j a_k\eta_l\alpha_i +a_i a_j\alpha_k \eta_l +a_i a_j\eta_k\alpha_l \Big) \xi^j \\ &- 2\rho \Big(a^t a_t \alpha_k \eta_l +a^t a_t \eta_k \alpha_l -a^ta^s \alpha_t \eta_s -a^ta^s \alpha_s \eta_t \Big)\alpha_i \end{align} In above the Einstein summation convention is used and $\rho$ is a real scalar.
The Problem is: I want to show this equation is not strictly parabolic.
I put $a=(1,0,\cdots ,0)$ and $\eta=(1,0,\cdots ,0))$. With this assumptions I only can show the first part is equal to zero.
Any suggestion is highly appreciated.