Let $M$ be a smooth manifold and $(\mathfrak{g}, [\cdot, \cdot])$ be a $\mathbb R$-Lie algebra. How can I introduce a Lie bracket on $\Gamma(TM\oplus (M\times \mathfrak{g}))$?
Above $TM\oplus (M\times \mathfrak{g})$ is the Whitney sum of the tangent bundle of $M$ and the trivial bundle $M\times \mathfrak{g}$.
I have some considerations for now: It is easy to see the map $\sigma:C^\infty(M)\otimes_{\mathbb R} \mathfrak{g}\longrightarrow \Gamma(M\times \mathfrak{g})$, $f\otimes v\longmapsto \sigma_{f, v}$ where $$\sigma_{f, v}(p)=(p, f(p)v),$$ is an isomorphism of $\mathbb R$-spaces. This way $$\Gamma(TM\oplus (M\times \mathfrak{g}))\simeq \Gamma(TM)\oplus (C^\infty(M)\otimes_{\mathbb R} \mathfrak{g}),$$ as $\mathbb R$-spaces. So, the attempt would be something like $$[(X, f\otimes u), (Y, g\otimes v)]=([X, Y], (X(g)-Y(f))\otimes [u, v]).$$ But it is not working because: $$\begin{align*} \displaystyle [(Y, g\otimes v), (X, f\otimes u)]&= ([Y, X], (Y(f)-X(g))\otimes [v, u])\\ &=(-[X, Y], -(X(g)-Y(f))\otimes (-[u, v]))\\ &=(-[X, Y], (X(g)-Y(f))\otimes [u, v]). \end{align*}$$ so it is not anti-symmetric. How can I fix this?