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 $W=\bigoplus\_{i\in I}(W\cap V\_i)$.
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> 


While this isn’t hard to prove (similar to Hoffman-Kunze 1971, [§7.5](//archive.org/details/LinearAlgebraHoffmanAndKunze/page/n270) 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](//books.google.com/books?id=11HxCAAAQBAJ&ots=CPps14aD40&pg=PA122&dq=%22Theorem+1%22)), and when using it (e.g. Lie groups [VIII.3.1](//books.google.com/books?id=oCO0xOzNLhAC&pg=PA86)) Bourbaki refers to such an abstrusely worded version (Algebra [VII.2.2](//books.google.com/books?id=GXT1CAAAQBAJ&ots=BSqgmQ9ECt&pg=PA253&dq=%22torsion+modules+over+a+principal+ideal+domain%22)) 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.



Invariant subspaces and isotypic decomposition (reference request)
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.


