Personally, I do not think that that proof is correct. This is a simple question of a compact homogeneous spaces. Any even dimensional compact Lie group is a (homogeneous) complex torus bundle over a projective rational homogeneous space (which is also simply connected---K"ahler-Einstein with positive Ricci curvature) and therefore is complex. The paper basically said that the complex structure J_H comes down to S^6 is integrable. His reason was that J_H is the restriction of J_{G_2} to H. However, H is not closed under the Lie bracket. That is why J_H can not simply come down to S^6.