Let $(X_i)_{i\in I}$ be an independent family of random variables on $(\Omega,\mathcal F,\mathbb P)$. Let $I_1,\dots,I_m\subset I$ be pairwise disjoint, and set
\begin{align*}
\mathcal G_r=\sigma(X_i:i\in I_r), \qquad r\in\{1,\dots,m\}.
\end{align*}
Then $\mathcal G_1,\dots,\mathcal G_m$ are independent sigma-algebras.