Let $M^5\simeq N^2\times k^3$ be a contact metric manifold and $\lbrace \xi, e_1,e_2=\varphi e_1, e_3,e_4=\varphi e_3 \rbrace$ is an $\varphi$-basis of smooth eigenvector $h=\dfrac{1}{2} \mathcal{L}_\xi \varphi$ on $M^5$.
$he_1=\lambda e_1$, $he_2=-\lambda e_2$, $he_3=\mu e_3$, $he_4=-\mu e_4$.
We know $N^2$ is locally symmetric and I showed $K^3$ is a semi-symmetric real cone and then it is conformally flat.
My question is this: How can I know which vectors in $\varphi$-basis is a basis for $K^3$?
I showed $e_1\in TK^3$. I am looking for two others one.