When constructing the Lie algebra $L(G)$ of a Lie group $G$, one usually uses the identification of the tangent space $T_1 G$ with left invariant vector fields $\mathcal{V}^l(G)$ to construct the Lie bracket on $L(G)=T_1 G$. However, in the literature on Lie groupoids and Lie algebroids some authors use left-invariant and some use right-invariant vector fields (and I think that right-invariant vector fields are more natural here).

There are several reasons for using one convention or the other, for instance using the right invariant vector fields yields the usual Lie bracket on the vector fields, which are the sections of the Lie algebroid of the pair groupoid. This can be rephrased by saying that the natural action of the bisections (which are just diffeomorphisms in the case of the pair groupoid) yield a Lie algebra homomorphism from the right invariant vector fields on the Lie groupoid to the vector fields on the manifold. However, in finite-dimensional Lie theory one often wants to make Lie algebra elements act on functions on the Lie group, which then involves a $^{-1}$ in the left regular representation and thus $\mathcal{V}^l(G)$ does act naturally on $C^\infty(G,\mathbb{R})$.

This has the unfortunate effect, that if we consider $G$ as a Lie groupoid with one object $(G\rightrightarrows *)$, then the Lie bracket on its Lie algebroid $L(G\rightrightarrows *)$ is not the same as the Lie bracket on the Lie algebra $L(G)$, considered as a Lie algebroid $L(G)\to *$.

Now I have two questions on this:

1. What are further reasons for using one or the other convention (for instance, calculations that are substantially easier in one or the other)?
2. In your opinion, is this is a historical accident that one should stick to and endure the resulting signs or do you see good reasons for braking with the conventions in favour of a unified construction of the Lie functor on Lie groups and Lie groupoids?

Note: One cannot simply use left-invariant vector fields on Lie algebroids, what is frequently done in the literature. One would also have to change the way how these objects act naturally, for instance that diffeomorphisms act on a manifold naturally from the left by evaluation.

When constructing the Lie algebra $L(G)$ of a Lie group $G$, one usually uses the identification of the tangent space $T_1 G$ with left invariant vector fields $\mathcal{V}^l(G)$ to construct the Lie bracket on $L(G)=T_1 G$. However, in the literature on Lie groupoids and Lie algebroids some authors use left-invariant and some use right-invariant vector fields (and I think that right-invariant vector fields are more natural here).

There are several reasons for using one convention or the other, for instance using the right invariant vector fields yields the usual Lie bracket on the vector fields, which are the sections of the Lie algebroid of the pair groupoid. This can be rephrased by saying that the natural action of the bisections (which are just diffeomorphisms in the case of the pair groupoid) yield a Lie algebra homomorphism from the right invariant vector fields on the Lie groupoid to the vector fields on the manifold. However, in finite-dimensional Lie theory one often wants to make Lie algebra elements act on functions on the Lie group, which then involves a $^{-1}$ in the left regular representation and thus $\mathcal{V}^l(G)$ does act naturally on $C^\infty(G,\mathbb{R})$.

This has the unfortunate effect, that if we consider $G$ as a Lie groupoid with one object $(G\rightrightarrows *)$, then the Lie bracket on its Lie algebroid $L(G\rightrightarrows *)$ is not the same as the Lie bracket on the Lie algebra $L(G)$, considered as a Lie algebroid $L(G)\to *$.

Now I have two questions on this:

1. What are further reasons for using one or the other convention (for instance, calculations that are substantially easier in one or the other)?
2. In your opinion, is this is a historical accident that one should stick to and endure the resulting signs or do you see good reasons for breaking with the conventions in favour of a unified construction of the Lie functor on Lie groups and Lie groupoids?

Note: One cannot simply use left-invariant vector fields on Lie algebroids, what is frequently done in the literature. One would also have to change the way how these objects act naturally, for instance that diffeomorphisms act on a manifold naturally from the left by evaluation.