The definition of an exact category as I have read is a pair ($\mathcal{C}, \mathcal{E}$) is an exact category if the additive category $\mathcal{C}$ is embedded as a full subcategory inside an abelian category $\mathcal{A}$, $\mathcal{E}$ is the collection of all sequence in $\mathcal{C}$ of the form $$0\rightarrow A_2\rightarrow A_1\rightarrow A_0\rightarrow 0$$ which are exact in $\mathcal{A}$ and $\mathcal{C}$ is closed under extension in $\mathcal{A}$.
Now Nil($R$) is the category that has objects as pairs ($P, \nu$) where $P$ is finitely generated projective $R$ module and $\nu$ is a nilpotent endomorphism of $P$.
Now I have been told that it is an exact category, but I am finding it difficult to establish an abelian category such that Nil($R$) embeds inside it as a full subcategory.
I understand that $P(R) $ the category of finitely generated projective $R$ module can be embedded in the category of $R$ modules and can be shown exact but as I have mentioned earlier I am finding the embedding part difficult to prove. Any hints or suggestion is highly appreciated.