What is the factor in the definition of [A,A] in differential forms?

[Dox]

Hi everyone, I've been studying a paper in which there is a connection given by,

$$A = f(r)\sigma_1 dx+g(r)\sigma_2 dy,$$​

where $$\sigma$$'s are half the Pauli matrices. I need to calculate the field strength,

$$F = dA +[A,A]$$.​

I have computed it, but a factor is given me problems. I would say,

$$dA = f' \sigma_1 dr\wedge dx + g'\sigma_2 dr\wedge dy$$​

and

$$[A,A] = 2 f g \sigma_3 dx\wedge dy,$$​

with a factor 2 coming from the fact that there are two contributions... like a binomial.

Is it OK or there is a half factor hidden in the definition of $$[A,A]$$?

Thank you so much.

DOX​

 

[martinbn]

There aren't two contributions. They are in the commutator of the sigmas.