Existence and uniqueness of a twisted one form which satisfies some inner product property Let $P\rightarrow M$ be a principal bundle over a (pseudo)-Riemannian manifold $(M,g)$, with compact structure group $G$ ($G$ is assumed to be a real Lie group), and a connection one form $A$. Let $W$ be a complex vector space, $\rho:G\rightarrow GL(W)$ a complex representation, and $\langle\cdot,\cdot\rangle_W$ be $G$ invariant Hermitian inner product on $W$. The associated vector $E=P\times_{\rho}W$ then carries an induced bundle metric given by:
$$
\begin{align}
\langle[p,w],[p,v]\rangle_E=\langle w,v\rangle_W
\end{align}
$$
which is well defined and independent of our choice of class representative. The pseudo Riemannian metric induces a scalar product of twisted one forms:
$$
\langle \omega,\eta\rangle_E=\langle \omega^i\otimes e_i,\eta^i\otimes e_j\rangle_E=\langle \omega^i,\eta^j\rangle\cdot\langle e_i,e_j \rangle_E
$$
where $\{e_i\}$ is a local frame for $E_U$,  $\omega^i,\eta^j\in \Omega^k(M,\mathbb{C})\cong \Omega^k(M)\otimes \mathbb{C}$, and $\langle\cdot,\cdot\rangle$ is the Hermitian scalar product on $\Omega^k(M,\mathbb{C})$
determined by $g$.
We also have a real vector bundle $\text{Ad}(P)=P\times_{\text{Ad}}\mathfrak{g}$, where $\text{Ad}$ is the adjoint representation of $G$ on $\mathfrak{g}$. Fixing a Ad-invariant
inner product on $\mathfrak{g}$ induces a bundle metric on $\text{Ad}(P)$. Furthermore,
for any $\alpha_M\in \Omega^1(M,\text{Ad}(P))$ and any section $\Phi$ of $E$, we can obtain a canonical one form $\alpha_M\cdot \Phi\in \Omega^1(M,E)$ which is defined in a local gauge as:
$$\alpha_M\cdot \Phi=[s, \rho_*(s^*\alpha)\phi]$$
for some $\phi:U\rightarrow W$, and the unique Ad-invariant, horizontal one form $\alpha\in \Omega^1(P,\mathfrak{g})$ satisfying
$$\alpha_M(X_x)=[p,\alpha_p(Y)]$$
for all $p$ and $Y$ such that $\pi(p)=x$ and $\pi_*Y=X_x\in T_xM$.
Let $d_A$ denote the exterior covariant derivative, I am trying to show that there exists a unique $J_H(A,\Phi)\in\Omega^1(M,\text{Ad}(P))$, such that for all $\alpha_M\in \Omega^1(M,\text{Ad}(P))$:
$$\langle \alpha_M,J_H(A,\Phi)\rangle_{\text{Ad}(P)}=2\Re(\langle d_A\Phi,\alpha_M\cdot \Phi\rangle_E)$$
where $\Re$ denotes taking the real part of a complex function. I think that existence follows from the nondegeneracy of both scalar products, but I can't figure out the quite argument...is there a clever way to argue this that is more clear than just appealing to non degeneracy, and calling it a day? Perhaps I am just unsure of why nondegeneracy implies existence...
Edit: Ok I can show uniqueness without going into coordinates as follows. Suppose $J_H$ exists, and is not unique, i.e. that there exists another form $\omega\in \Omega^1(M,\text{Ad}(P))$ such that for all $\alpha_M\in \Omega^1(M,\text{Ad}(P))$:
$$\langle \alpha_M,\omega\rangle_{\text{Ad}(P)}=\langle d_A\Phi,\alpha_M\cdot \Phi\rangle$$
Then we have that:
$$\begin{align}
\langle \alpha_M,J_H-\omega\rangle_{\text{Ad}(P)}=&
\langle \alpha_M,J_H\rangle_{\text{Ad}(P)}-\langle \alpha_M,\omega\rangle_{\text{Ad}(P)}\\
=&2\Re(\langle d_A\Phi,\alpha_M\cdot \Phi\rangle_E)-2\Re(\langle d_A\Phi,\alpha_M\cdot \Phi\rangle_E)\\
=&0
\end{align}$$
for all $\alpha_M\in \Omega^1(M,\text{Ad}(P))$ but $\langle\cdot,\cdot\rangle_{\text{Ad}(P)}$ is non degenerate, so we have a contradiction and $J_H$ is unique. The question is now how do I show existence? I'm also going to delete the work in coordinates above as it is no longer helpful to the discussion, is kinda a lot.
 A: Ok, nondegeneracy implies existence as follows:
Note that:
$$\begin{align*}
    2\Re(\langle d_A\Phi,\alpha_M\cdot \Phi\rangle_E)=&\langle d_A\Phi,\alpha_M\cdot \Phi
    \rangle_E+\langle\alpha_M\cdot \Phi,d_A\Phi\rangle_E
\end{align*}$$
Furthermore, the assignment:
$$\begin{align*}
    \Lambda:\Omega^1(M,\text{Ad}(P))&\longrightarrow C^\infty(M)\\
    \alpha_M&\longmapsto \langle d_A\Phi,\alpha_M\cdot \Phi
    \rangle_E+\langle\alpha_M\cdot \Phi,d_A\Phi\rangle_E
\end{align*}$$
is clearly a $C^\infty(M)$ linear map, thus $\Lambda$ is a global section of
$TM\otimes \text{Ad}(P)^*$, where $\text{Ad}(P)^*$ is bundle dual to $\text{Ad}(P)$. Since the
bundle metric $\langle \cdot,\cdot\rangle_{\text{Ad}(P)}$ on $T^*M\otimes \text{Ad}(P)$ is non
degenerate, it follows that it induces a bundle isomorphism:
$$\begin{align*}
    F:T^*M\otimes \text{Ad}(P)\longrightarrow TM\otimes \text{Ad}(P)^*
\end{align*}$$
that satisfies:
$$\begin{align*}
    F(\omega)(\eta)=\langle \omega,\eta\rangle_{\text{Ad}(P)}
\end{align*}$$
for all $\omega,\eta\in \Omega^1(M,\text{Ad}(P))$. Setting $J_H=F^{-1}(\Lambda)$, then implies
the claim as for all $\alpha_M\in\Omega^1(M,\text{Ad}(P))$:
$$\begin{align*}
    \langle J_H,\alpha_M\rangle_{\text{Ad}(P)}=&F(J_H)(\alpha_M)\\
    =&\Lambda(\alpha_M)\\
    =&2\Re(\langle d_A\Phi,\alpha_M\cdot \Phi\rangle_E)
\end{align*}$$
