How is the connection 1 form defined?

Question

I am studying connection on principal bundles and the definition I've encountered is the following "A connection over a principal G-bundle p:P$\rightarrow$B is a splitting of the Altiah sequence $$\tilde{ω}.TP\rightarrow VP ≃ P\times\mathfrak{g}$$ $$u\mapsto (\pi_{P}(u),ω|_{\pi_{p}(u)}(u))$$ where $ω$ is a 1-form over P with values in $\mathfrak{g}$, $ω\in \mathcal{A}^{1}(P)\otimes\mathfrak{g}$ that is G-equivariant". Now, I can't understand how the latter bundle is defined. Cause we know how to build a tensor product of two vector bundles, but here $\mathfrak{g}$ is just $T_{e}G$ (a vector space), so which bundle should I consider?