Is it possible to combine two Sobel kernels into one?
A Sobel edge detector compares an approximation of the image gradient $|\nabla f_c(x,y)|$ to a threshold to decide if a pixel is an edge or not. A proper threshold must be determined and computed so that the comparison produces useful results. The approximation is given by the following:
$$|\nabla f_c(x,y)| \longrightarrow \sqrt{ (f \star h_x)^2 + (f \star h_y)^2 }$$
where $f = f[n_1,n_2]$ is the discrete-time image sequence, and $h_x = h_x[n_1,n_2]$ and $h_y = h_y[n_1,n_2]$ are the two directional Sobel kernels respectively.
As seen, the Sobel operator is nonlinear in the filters $h_x$ and $h_y$, and a single kernel, which would replace those two directional kernels, cannot be obtained in this case.