Let $G_2$ denote the exceptional simple Lie group of 14 dimension and $SU(3)\subset G_2$. Consider $S^6\cong G_2/SU(3)$.
$\textbf{Q1:}$ How do I see above isomorphism?
$\textbf{Q2:}$ "From definition of $G_2$ and its structure equations one sees immediately that $S^6$ has a non-integrable almost complex structure." The commutator relation of complexified lie algebra indicates the tangent vector field commutator relation. However, to get non-integrable almost complex structure, one needs complex coefficient 1 forms and its exterior derivatives. Almost complex structure only gives a splitting of 1-forms into holomorphic and anti-holomorphic 1 forms. How do I see there is indeed non-integrable almost complex structure? Is this something following directly from commutator relations?
Ref. Chern, Shiing-shen, Complex manifolds without potential theory pg 16 last paragraph.