Equivariant cohomology, identity

Question

I'm studying equivariant cohomology on three references:

1. Szabo's review about equivariant localization (S);
2. Libine's note on equivariant cohomology (L);
3. Berline, Getzler, Vigne's book "Heat Kernels and Dirac Operators", Springer (BGV).

I'm deriving all formulas by myself. I face two problems:

1. In (S) when he derive (2.83) he uses a commutator while (L), in the longest formula below Definition 34 writes

$$ i_X\nabla + \nabla i_X. $$

I agree with (L) because it's just the square:

$$ (\nabla - i_X)^2, $$

so why (S) put the commutator? (BGV) seems to agree with (S).

2. Then I compute

$$ \begin{align} (i_X\nabla + \nabla i_X)\alpha & = i_X(d\alpha + \omega\wedge\alpha) + (d + \omega)i_X\alpha\\ & = i_Xd\alpha + i_X\omega\wedge\alpha - \omega\wedge i_X\alpha + di_X\alpha + \omega\wedge i_X\alpha\\ & = (i_Xd\alpha + \omega(X)\alpha) +di_X\alpha\\ & = \nabla_X\alpha + di_X\alpha \end{align}. $$

So I get an extra $di_X\alpha$. This must be correct because also (BGV) write (page 211)

$$ [\nabla,i_X]=\nabla_X $$

which is, apart for the mysterious commutator instead anticommutator, the same formula.