[proofplan]
We first differentiate the $G$-invariance of $B$ to obtain the infinitesimal skew-adjointness identity for the Lie algebra action $\rho_*: \mathfrak g \to \mathfrak{gl}(V)$. Then we compute the desired identity in a local trivialization determined by a local section of the principal bundle $\pi:P\to M$ and the principal connection form $\omega\in\Omega^1(P;\mathfrak g)$. In the associated-bundle convention $[pg,v]=[p,\rho(g)v]$, the induced connection is the ordinary derivative corrected by the local connection form with a plus sign, and the infinitesimal invariance identity cancels exactly the two correction terms. Since equality of $1$-forms can be checked locally, the local computation proves the global statement.
[/proofplan]
[step:Differentiate the $G$-invariance of $B$ to obtain infinitesimal skew-adjointness]
Let
\begin{align*}
\rho_* := d\rho_e: \mathfrak g \to \mathfrak{gl}(V)
\end{align*}
denote the differential of $\rho$ at the identity element $e \in G$.
We claim that for every $\xi \in \mathfrak g$ and every $v,w \in V$,
\begin{align*}
B(\rho_*(\xi)v,w) + B(v,\rho_*(\xi)w) = 0.
\end{align*}
To prove this, fix $\xi \in \mathfrak g$ and let
\begin{align*}
\gamma: (-\varepsilon,\varepsilon) \to G
\end{align*}
be a smooth curve satisfying $\gamma(0)=e$ and $\gamma'(0)=\xi$. For fixed $v,w \in V$, define $F: (-\varepsilon,\varepsilon) \to \mathbb R$ by
\begin{align*}
F(r) = B(\rho(\gamma(r))v,\rho(\gamma(r))w).
\end{align*}
By the $G$-invariance of $B$, the function $F$ is constant and equal to $B(v,w)$. Hence $F'(0)=0$. Differentiating at $r=0$ using bilinearity of $B$ gives
\begin{align*}
0 = B(\rho_*(\xi)v,w) + B(v,\rho_*(\xi)w).
\end{align*}
This proves the infinitesimal identity.
[/step]
[step:Write the associated bilinear form and connection in a local trivialization]
Let $M$ denote the smooth base manifold, let $\pi:P\to M$ denote the principal $G$-bundle projection, and let $\omega\in\Omega^1(P;\mathfrak g)$ denote the fixed principal connection form. Fix an [open set](/page/Open%20Set) $U \subset M$ and a smooth local section
\begin{align*}
\sigma: U \to P
\end{align*}
of $\pi: P \to M$. Let
\begin{align*}
A := \sigma^*\omega \in \Omega^1(U;\mathfrak g)
\end{align*}
be the local connection form determined by $\sigma$.
Let $s,t \in \Gamma(E)$. Over $U$, there are unique smooth maps $u: U \to V$ and $v: U \to V$ such that
\begin{align*}
s(x) = [\sigma(x),u(x)]
\end{align*}
and
\begin{align*}
t(x) = [\sigma(x),v(x)]
\end{align*}
for every $x \in U$.
In this local trivialization, the definition of $B_E$ gives
\begin{align*}
B_E(s,t)(x) = B(u(x),v(x)).
\end{align*}
Let $\mathfrak X(U) := \Gamma(TU)$ denote the space of smooth vector fields on $U$. For a smooth vector field
\begin{align*}
X \in \mathfrak X(U),
\end{align*}
using the convention $[\sigma(x)g,a]=[\sigma(x),\rho(g)a]$ for the associated bundle, the induced connection on the associated bundle is represented by
\begin{align*}
(\nabla_X s)(x) = [\sigma(x), X(u)(x) + \rho_*(A(X)_x)u(x)]
\end{align*}
and
\begin{align*}
(\nabla_X t)(x) = [\sigma(x), X(v)(x) + \rho_*(A(X)_x)v(x)].
\end{align*}
Here $A(X): U \to \mathfrak g$ is the smooth map obtained by evaluating the $\mathfrak g$-valued $1$-form $A$ on $X$, and $X(u): U \to V$ denotes the directional derivative of the smooth map $u: U \to V$ along $X$.
[/step]
[step:Compute the derivative of $B_E(s,t)$ and cancel the connection terms]
Fix $x \in U$ and $X \in \mathfrak X(U)$. Using the local expression
\begin{align*}
B_E(s,t)(x) = B(u(x),v(x)),
\end{align*}
the ordinary product rule for the fixed bilinear map $B: V \times V \to \mathbb R$ gives
\begin{align*}
d(B_E(s,t))_x(X_x) = B(X(u)(x),v(x)) + B(u(x),X(v)(x)).
\end{align*}
On the other hand, using the local formula for the induced connection,
\begin{align*}
B_E(\nabla_X s,t)(x) = B(X(u)(x) + \rho_*(A(X)_x)u(x),v(x))
\end{align*}
and
\begin{align*}
B_E(s,\nabla_X t)(x) = B(u(x),X(v)(x) + \rho_*(A(X)_x)v(x)).
\end{align*}
Adding these two identities and using bilinearity of $B$ gives
\begin{align*}
B_E(\nabla_X s,t)(x) + B_E(s,\nabla_X t)(x) = B(X(u)(x),v(x)) + B(u(x),X(v)(x)) + B(\rho_*(A(X)_x)u(x),v(x)) + B(u(x),\rho_*(A(X)_x)v(x)).
\end{align*}
Apply the infinitesimal identity from the first step with
\begin{align*}
\xi := A(X)_x \in \mathfrak g,
\end{align*}
with $u(x) \in V$ in place of $v$, and with $v(x) \in V$ in place of $w$. It gives
\begin{align*}
B(\rho_*(A(X)_x)u(x),v(x)) + B(u(x),\rho_*(A(X)_x)v(x)) = 0.
\end{align*}
Therefore
\begin{align*}
B_E(\nabla_X s,t)(x) + B_E(s,\nabla_X t)(x) = B(X(u)(x),v(x)) + B(u(x),X(v)(x)).
\end{align*}
Comparing with the derivative computed above yields
\begin{align*}
d(B_E(s,t))_x(X_x) = B_E(\nabla_X s,t)(x) + B_E(s,\nabla_X t)(x).
\end{align*}
[guided]
We now prove the key local identity in full detail. The point of choosing the local section $\sigma: U \to P$ is that it turns sections of the associated bundle into ordinary $V$-valued functions. Thus the sections $s,t \in \Gamma(E)$ are represented over $U$ by smooth maps
\begin{align*}
u: U \to V
\end{align*}
and
\begin{align*}
v: U \to V
\end{align*}
through the formulas
\begin{align*}
s(x) = [\sigma(x),u(x)]
\end{align*}
and
\begin{align*}
t(x) = [\sigma(x),v(x)].
\end{align*}
In this trivialization the associated [bilinear form](/page/Bilinear%20Form) is exactly the original bilinear form applied fibrewise:
\begin{align*}
B_E(s,t)(x) = B(u(x),v(x)).
\end{align*}
Therefore, for a smooth vector field $X \in \mathfrak X(U)$ and a point $x \in U$, the product rule for the fixed bilinear map $B: V \times V \to \mathbb R$ gives
\begin{align*}
d(B_E(s,t))_x(X_x) = B(X(u)(x),v(x)) + B(u(x),X(v)(x)).
\end{align*}
The connection introduces correction terms. The principal connection form is the fixed form $\omega\in\Omega^1(P;\mathfrak g)$, and the local connection form determined by the section $\sigma$ is
\begin{align*}
A := \sigma^*\omega \in \Omega^1(U;\mathfrak g).
\end{align*}
Evaluating $A$ on the vector field $X$ gives a smooth map
\begin{align*}
A(X): U \to \mathfrak g.
\end{align*}
With the associated-bundle convention $[\sigma(x)g,a]=[\sigma(x),\rho(g)a]$, the associated connection is represented locally by
\begin{align*}
(\nabla_X s)(x) = [\sigma(x), X(u)(x) + \rho_*(A(X)_x)u(x)]
\end{align*}
and
\begin{align*}
(\nabla_X t)(x) = [\sigma(x), X(v)(x) + \rho_*(A(X)_x)v(x)].
\end{align*}
Applying $B_E$ to these expressions gives
\begin{align*}
B_E(\nabla_X s,t)(x) = B(X(u)(x) + \rho_*(A(X)_x)u(x),v(x))
\end{align*}
and
\begin{align*}
B_E(s,\nabla_X t)(x) = B(u(x),X(v)(x) + \rho_*(A(X)_x)v(x)).
\end{align*}
Now expand both terms using bilinearity:
\begin{align*}
B_E(\nabla_X s,t)(x) + B_E(s,\nabla_X t)(x) = B(X(u)(x),v(x)) + B(u(x),X(v)(x)) + B(\rho_*(A(X)_x)u(x),v(x)) + B(u(x),\rho_*(A(X)_x)v(x)).
\end{align*}
The last two terms are exactly where the preservation of $B$ by the structure group is used, so we derive the needed infinitesimal identity inside this guided argument. Fix $\xi \in \mathfrak g$ and $a,b \in V$. Choose a smooth curve
\begin{align*}
\gamma: (-\varepsilon,\varepsilon) \to G
\end{align*}
satisfying $\gamma(0)=e$ and $\gamma'(0)=\xi$. Define
\begin{align*}
F: (-\varepsilon,\varepsilon) \to \mathbb R
\end{align*}
by
\begin{align*}
F(r) = B(\rho(\gamma(r))a,\rho(\gamma(r))b).
\end{align*}
Because $B$ is preserved by $\rho(G)$, we have $F(r)=B(a,b)$ for every $r \in (-\varepsilon,\varepsilon)$, so $F'(0)=0$. Differentiating at $r=0$ using the ordinary product rule for the fixed bilinear map $B$ and the definition $\rho_* = d\rho_e$ gives
\begin{align*}
0 = B(\rho_*(\xi)a,b) + B(a,\rho_*(\xi)b).
\end{align*}
Thus every $\xi \in \mathfrak g$ satisfies
\begin{align*}
B(\rho_*(\xi)a,b) + B(a,\rho_*(\xi)b) = 0
\end{align*}
for every $a,b \in V$. We apply this with
\begin{align*}
\xi := A(X)_x,
\end{align*}
with $a := u(x)$, and with $b := v(x)$. This gives
\begin{align*}
B(\rho_*(A(X)_x)u(x),v(x)) + B(u(x),\rho_*(A(X)_x)v(x)) = 0.
\end{align*}
Thus the two connection correction terms cancel, leaving
\begin{align*}
B_E(\nabla_X s,t)(x) + B_E(s,\nabla_X t)(x) = B(X(u)(x),v(x)) + B(u(x),X(v)(x)).
\end{align*}
This is precisely the expression obtained for $d(B_E(s,t))_x(X_x)$, so
\begin{align*}
d(B_E(s,t))_x(X_x) = B_E(\nabla_X s,t)(x) + B_E(s,\nabla_X t)(x).
\end{align*}
[/guided]
[/step]
[step:Pass from the local computation to the global identity]
The preceding computation holds on every open set $U \subset M$ admitting a smooth local section $\sigma: U \to P$, for every $x \in U$, and for every tangent vector $X_x \in T_xM$ represented by a local vector field $X \in \mathfrak X(U)$. Principal bundles admit such local sections around every point of $M$, and on each such trivializing open set the preceding formulas are exactly the standard local expressions for associated bundles and their connections under the stated convention.
Both sides of
\begin{align*}
d(B_E(s,t)) = B_E(\nabla s,t) + B_E(s,\nabla t)
\end{align*}
are globally defined smooth $1$-forms on $M$. Since the two $1$-forms agree on each member of a local trivializing cover, they agree on all of $M$. This proves the stated metric compatibility identity for every $s,t \in \Gamma(E)$.
[/step]
