What is the unity matrix in cylindrical coordinates?

Question

If $e_{\rho},e_{\varphi},e_z$ are the unit vectors in cylindrical coordinates, one can check \begin{align} (e_{\rho}\otimes e_{\rho})\cdot (e_{\rho}\otimes e_{\rho})= e_{\rho}\otimes e_{\rho}, (e_{\varphi}\otimes e_{\varphi})\cdot (e_{\varphi}\otimes e_{\varphi})= e_{\varphi}\otimes e_{\varphi}, (e_{z}\otimes e_{z})\cdot (e_{z}\otimes e_{z})= e_{z}\otimes e_{z}. \end{align} which means that the matrices are orthogonal. I suspect that $I=e_{\rho}\otimes e_{\rho}+e_{\varphi}\otimes e_{\varphi}+e_z\otimes e_z$. At least if one multiplies by a vector one gets back the vector. Is my conjecture true?