What's the isomorphism $Ω^{1}(B)\otimes\mathfrak{g}\sim Ω^{1}(B,ad(P))$?

Question

Let $p:P\rightarrow B$ be a principal $G$-bundle over a smooth manifold $B$. How can I show that $$Ω^{1}(B)\otimes\mathfrak{g} ≃ Ω^{1}(B,ad(P))$$ where $Ω^{1}(B,ad(P))\equiv\Gamma(\Lambda^{k}(T^{*}B)\otimes ad(P))=Hom(\Lambda^{k}(T^{*}B),ad(P))$?

Here I used that $ad(P):=P\times_{G}\mathfrak{g}$.