TITLE: Invariant subspaces and isotypic decomposition (reference request)
QUESTION [5 upvotes]: If $G$ is a (say) compact group and $V=\bigoplus_{i\in I}V_i$ the isotypic (a.k.a. primary) decomposition of a $G$-module, then 

any $G$-invariant subspace $W\subset V$ writes $W=\bigoplus_{i\in I}(W\cap V_i)$.

While this isn’t hard to prove (similar to Hoffman-Kunze 1971, §7.5 for a single operator), it seems silly to redo it in a paper. Unfortunately, the only reference I’m familiar with omits the proof (Kirillov 1976, §8.3), and when using it (e.g. Lie groups VIII.3.1) Bourbaki refers to such an abstrusely worded version (Algebra VII.2.2) that unpacking it takes as much work as a direct proof.
Q: What is a good reference to quote for this? Bonus points if the case of non-algebraically closed fields is spelled out.

REPLY [2 votes]: As suggested by the OP, I am turning my comment into an answer: a better Bourbaki reference is Algèbre VIII (new edition), §4, Proposition 4 d) (unfortunately not yet translated, as far as I know). It works for semi-simple modules over an arbitrary ring.