I am trying to prove that the Euler equation for a Riemannian 3-manifold $M$ $$ \frac{\partial u}{\partial t} + \nabla_{u} u = - \operatorname{grad} p \qquad \text (1) $$ is equivalent to $$ \frac{\partial u^\flat}{\partial t} + \mathcal L_u u^\flat = \frac{1}{2} d(u^\flat(u)) - dp \qquad \text (2) $$ where $u$ is a (velocity) vector field, $p$ is a function on $M$ (pressure term), $\mathcal L_u$ is the Lie derivative along $u$, and $\flat$ is the musical isomorphism from the tangent bundle to the cotangent bundle.
By applying $\flat$ to (1) and recalling that $\operatorname{grad} p = (dp)^\sharp$ one easily arrives to $$ \frac{\partial u^\flat}{\partial t} + (\nabla_{u} u)^\flat = - dp \qquad \text (3) $$ which is close already! Now the only step remaining is to show that \begin{equation} (\nabla_{u} u)^\flat = \mathcal L_u u^\flat - \frac{1}{2} d(u^\flat(u)) \qquad (4) \end{equation}
I have arrived to the following local expression for the LHS in terms of the metric $(g_{ij})$, a local frame $(\partial_i)$ and its coframe $(dx^i)$, thus considering $u=u^i \partial_i$: $$ (\nabla_{u} u)^\flat = g_{lm} \left( u^j\partial_j(u^m) + u^i u^j \Gamma^m_{ij} \right) dx^l \qquad \text (5) $$
My doubt is how can I arrive to the same expression for the RHS of (4). Thanks a lot for your help!