Gram-Schmidt orthogonalization process with specific dot product

Question

I have three vectors $$v_1=(1,1,1)^T$$ $$v_2=(1,1,0)^T$$ $$v_3=(1,0,0)^T$$ and special dot product definition $$(\overline{(x_1,x_2,x_3)},\overline{(y_1,y_2,y_3)})=2x_1y_2+x_1y_1+2x_2y_1+x_3y_3$$ I use standard Gram-Schmidt orthogonalization process. So $$e_1=v_1=(1,1,1)^T$$ $$e_2 = v_2 - \frac{(v_2, e_1)}{(e_1, e_1)} \cdot e_1 = (1,1,-5)^T$$ and $$e_3 = v_3 - \frac{(v_3,e_1)}{(e_1,e_1)}\cdot e_1 - \frac{(v_3,e_2)}{(e_2,e_2)} \cdot e_2 = (2,-3,0)^T$$ But $(e_2, e_3) = -9$ instead of $0$. I can't recognize my mistake. Thank you in advance!