Any given representation (some matrices of some algebra) will be reducible if there exists a singular, but nonzero matrix S that commutes with all elements of the representation; conversely, if no such S exists, then the representation is irreducible. This is Schur's Lemma.

Question 1: Does there exists some analogously simple criterion for deciding whether or not a given representation is decomposable, i.e., whether or not there exists some change of basis (i.e., a similarity transformation) that brings the representation on (sub)block-diagonal form?

Only very recently I learned that the two concepts are not equivalent; as a physicist I have always thought so. A concrete example is "... the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices" [http://en.wikipedia.org/wiki/Irreducible_representation].

Question 2: When are the two concepts equivalent?

Any given representation (some matrices of some algebra) will be reducible if there exists a singular, but nonzero matrix S that commutes with all elements of the representation; conversely, if no such S exists, then the representation is irreducible. This is Schur's Lemma.

Question 1: Does there exists some analogously simple criterion for deciding whether or not a given representation is decomposable, i.e., whether or not there exists some change of basis (i.e., a similarity transformation) that brings the representation on (sub)block-diagonal form?

Only very recently I learned that the two concepts are not equivalent; as a physicist I have always thought so. A concrete example is "... the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices" [http://en.wikipedia.org/wiki/Irreducible_representation].

Question 2: When are the two concepts equivalent? I expect that they are so at least for the specific case of finite-dimensional, unitary representations (or am I mistaken?), but are there other cases?