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.