Let $\mathfrak{g}$ be a finite dimensiinal Lie algebra. Let $e^i$ be a basis of $\mathfrak{g}$, $e_i \in \mathfrak{g}^*$ the dual basis. We denote by $x_i$ the coordinate functions w.r.t this basis.
In the proof of Lemma 25 of the article of Equivariant cohomology with generalized coefficients" the authors say
For $X \in \mathfrak{g},$ denote by $\partial_X$ the constant coefficient vector field on $\mathfrak{g}$ equal to $X$. The vector field Y:= $\sum_{i,j}x_ix_j \partial_{[e^i,e^j]}$ is the vector field equal at the point $X \in \mathfrak{g}$ to $[X,X]=0$.
I don't understand why is the vector field $Y$ equal to zero ?
$Y$ is not 0. It is zero at the point $X$. To see this, write $X$ as a linear combination of the basis elements $e^i$, and compute the finite sum. Keep in mind the definition of the commutator, in local coordinates.