= Action summand can fail for an infinite-dimensional simple module
Take the <Verma module> for the <sl2 Lie algebra> of highest weight $-2$, with basis $v_m$ and $fv_m=v_{m+1}$, $hv_m=(-2-2m)v_m$, $ev_m=-m(m+1)v_{m-1}$. It is simple. In $\mathfrak{sl}_2\otimes V$, the highest vectors of weight $-2$ are multiples of $w=e\otimes v_1-h\otimes v_0=f(e\otimes v_0)$. Every homomorphism to $V$ kills $e\otimes v_0$, whose weight is zero, and hence kills $w$. Every embedded copy of $V$ would have its highest vector proportional to $w$, so no projection onto such a copy can exist. Thus semisimplicity of the Lie algebra alone does not provide an action summand in infinite dimension.
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