Is there any relation between Lie algebra being simple and its representation being irreducible?
---- sorry for being vague; here's some detail about this question:
-- If $g$ is simple, then $g$ can certainly have decomposable representations, for instance, just take two representations on $V_1$ and $V_2$, and we have a decomposable representation on $V_1\oplus V_2$.
-- On the other hand, if $g$ is not simple, say $g = g_1 \oplus g_2$, but $g$ has an irreducible representation on $V$, then, is it true that either $g_1$ acts on $V$ trivially or $g_2$ acts on $V$ trivially?