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This paper investigates a universal PBW-basis and a minimal set of generators for the Hall algebra $\mathscr{H}\left(C_2\left(𝒫\right)\right)$, where $C_2\left(𝒫\right)$ is the category of morphisms between projective objects in a finitary hereditary exact category $𝒜$. When $𝒜$ is the representation category of a Dynkin quiver, we develop multiplication formulas for the degenerate Hall Lie algebra $ℒ$, which is spanned by isoclasses of indecomposable objects in $C_2\left(𝒫\right)$. As applications, we demonstrate that $ℒ$ contains a Lie subalgebra isomorphic to the central extension...