Solution
= Solution
Direct differentiation gives, for example,
$$
d\sigma^3=-\sin\theta\,d\theta\wedge d\phi
=\sigma^1\wedge\sigma^2.
$$
Differentiating $\sigma^1,\sigma^2$ similarly and substituting the definitions yields $d\sigma^1=\sigma^2\wedge\sigma^3$ and $d\sigma^2=\sigma^3\wedge\sigma^1$. Hence these <Maurer-Cartan form>[Maurer-Cartan forms] on the three-sphere obey
$$
\boxed{d\sigma^i=\frac12\delta^{il}\epsilon_{ljk}
\sigma^j\wedge\sigma^k}.
$$