homework and exercises - Double line notation - gluon propagator

Consider the following Feynman diagram for a gluon (there is momentum $k$ flowing between the two points):

Normally, the factor that goes along with this is $G^{ab}_{\mu\nu}(k) \ = \ \delta_{ab} \frac{- i }{ k^{2} - i \epsilon_{+} } \left( \eta_{\mu \nu} + ( \xi - 1 ) \frac{ k_{\mu} k_{\nu} }{ k^{2} } \right)$.

Using t'Hooft's double line formalism, the above diagram can be drawn as:

My task is to give a guess as to what the corresponding factor should be. I $think$ that it's

$$G^{(\bar{i}, j)(\bar{l},k)}_{\mu\nu}(k) \ = \ \delta_{jl} \delta_{ik} \frac{- i }{ k^{2} - i \epsilon_{+} } \left( \eta_{\mu \nu} + ( \xi - 1 ) \frac{ k_{\mu} k_{\nu} }{ k^{2} } \right)$$

However, this seems a little too simple and I'm worried that I'm missing out on a detail. When I google the "t'Hooft double line formalism" I find myself a lot of material, however I find that the discussion on the vertex factors is lacking (the same goes for the three- and four- gauge boson vertices). Can somebody either help me understand if I have the correct factor or point me in the direction of some literature that can?

(I should mention that the parameter $\xi$ describes the gauge we're working in)

EDIT: I forgot to say that I'm talking about the $U(N)$ gauge group.