Let $(M,\xi)$ be a contact 3 manifold and $\eta\in \Omega^1(M)$, $R\in \mathfrak X(M)$ be the relevant 1-form (with $\xi=Ker(\eta)$) and the Reeb vector field. In view of the splitting $TM=\xi\oplus \langle R\rangle$, I want to construct a metric on $M$ given by $g=g_h+g_v$ where $g_h(-,-)=d\eta(-,J-)$ is a metric on $\xi$ for some almost complex structure $J:\xi\to \xi$ and $g_v$ is a metric on $\langle R\rangle$ probably given by the symmetric product $\eta \odot\eta$.
Can someone guide me through the proof (or a reference) of the existence of such a nicely adapted metric? Such $J$ is probably constructed fiberwise at first and then it should be possible to construct a bundle of all the $J_p$, with contractible fibers.
Regarding the vertical metric, how does one define a positive definite tensor from the symmetric product $\eta^2$?