There are many versions of the Cochran theorem and i don't know how to link the versions.
My version is :
Let, for $j \in \{1,..,p\}$, $Q_j \in S_n(\mathbb{R})$ with ranks $r_1,...,r_j$ respectively and $Id_n=\sum_{i=1}^p Q_i$. Let $q_j(x)=^txQ_jx$ the quadratics forms. Let ($X_i$) iid with $X_i \sim N(m_i,1)$, $M=(m_1,...,m_n)$ and the gaussian vector $X \sim N_n(M,Id_n)$.
If $\sum_{i=1}^p r_i=n$, so $q_j(X)$ are independent and $q_j(X) \sim \chi^2(r_j,\lambda_j)$ (Noncentral chi-squared distribution) with $\lambda_j=q_i(M)$.
The most common version is :
Let $X$ a gaussian vector ($ X \sim N(0,Id)$) and $F$ a linear subspace of $\mathbb{R}^n$ with dimension $d$. Let $P_F$ and $P_F^\perp$ the orthogonal projection over $F$ and $F^\perp$.
1) $P_F$ and $P_F^\perp$ are independent and $P_F \sim N(0,P_F)$ ,$P_F^\perp \sim N(0,P_F^\perp)$
2) $||P_F||^2$ and $||P_F^\perp||^2$ are independent and $||P_F||^2 \sim \chi^2(d)$, $||P_F^\perp||^2 \sim \chi^2(n-d)$
Cordially, doeup