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To calculate A (1 − g , 2 + g , 3 − h , 4 + $_{h}$), we need six-point non-MHV tree amplitudes for gluons A (1 − g , 6 + g , 3 − g , 5 + g , 4 − g , 2 + g ) and A (1 − g , 3 − g , 5 + g , 4 − g , 2 + g , 6 + g ). Using the tree amplitude with six gluons given in[25] and substituting k$_{3}$ , k$_{4}$ and k$_{5}$ , k$_{6}$ by k$_{3}$ 2 and k$_{4}$ 2 correspondingly, we get 

<img src='content_image/297197.jpg'>

Then we substitute the amplitudes(13) into the relation(6c). The amplitude A (1 − g , 2 + g , 3 − h , 4 + $_{h}$) is given 

<img src='content_image/297198.jpg'>

So far, we have given all the independent three- and four-point amplitudes, other three- and four-point amplitudes can be derived from these amplitudes by using a parity transfor- mation or performing an appropriate replacement on the external legs. These results are same with those given by KLT relations[17]. Then the three- and four-point tree ampli- tudes for gluons minimally coupled to graviton satisfy D$_{2}$ relations. We then expect the D$_{2}$ relations hold in all amplitudes where gluons are minimally coupled to gravitons. 

