= Nonsplit zero-weight Verma extension for sl2
{title2=$0\to M(-2)\to M(0)\to\mathbb C\to0$}
The <Verma module> $M(0)$ has basis $v_n=f^nv_0$, with $hv_n=-2nv_n$, $fv_n=v_{n+1}$ and $ev_n=n(1-n)v_{n-1}$. Its submodule spanned by $v_n$ for $n\geq1$ is $M(-2)$, and the quotient is the trivial one-dimensional module. A complementary submodule would be trivial and hence lie in the zero-weight space $\mathbb Cv_0$, but $fv_0=v_1\ne0$. Therefore the displayed extension does not split. This explains why the finite-dimensional hypothesis in the <Weyl complete reducibility theorem> is essential.
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