= Solution
{c}
Use the corrected <Lie algebra representation> $E=-e$, $H=h$, $F=f$ from part (i). A <vector subspace> is invariant under $E,H,F$ exactly when it is invariant under the printed $e,h,f$, since changing the sign of a <linear map> does not change its <invariant subspaces>. Define
$$
W=\bigoplus_{i\geq1}kv_i.
$$
This is a nonzero proper <submodule>: $H$ preserves each $kv_i$, $F$ increases the index, $E(v_1)=0$, and $E(v_i)=-(i-1)v_{i-1}\in W$ for $i\geq2$.
We must rule out <submodules> consisting only of mixtures of <weight vectors>. Let $U\ne0$ be any <submodule>, and choose $u=\sum_{j\in S}a_jv_j\in U$ with $S$ finite and every $a_j\ne0$. The <eigenvalues> $-2j$ of $H$ are distinct. For $i\in S$, the <Lagrange interpolation polynomial>
$$
p_i(t)=\prod_{\substack{j\in S\\j\ne i}}\frac{t+2j}{-2i+2j}
$$
satisfies $p_i(H)u=a_iv_i$. Since $U$ is invariant under $H$, it is invariant under every <polynomial> in $H$, so $v_i\in U$.
If $i=0$, repeated applications of $F$ give $F^mv_0=m!v_m$, and $U=V$. If $i\geq1$, repeated applications of $E$ give a nonzero multiple of $v_1$, because the successive coefficients $-(i-1),\ldots,-1$ are nonzero in <characteristic zero>. Repeated applications of $F$ to $v_1$ then give every $v_j$ with $j\geq1$, so $W\subseteq U$.
Finally, if such a $U$ contains any vector outside $W$, that vector has a nonzero coefficient of $v_0$; the same <polynomial interpolation> extracts $v_0$, forcing $U=V$. Hence the complete list of <submodules> is $0,W,V$. In particular, $W$ is a <simple module>. The <quotient module> $V/W$ has <basis> $\overline v_0$; $E\overline v_0=H\overline v_0=0$ and $F\overline v_0=\overline v_1=0$. It is the one-dimensional <trivial Lie algebra representation>.
The <module extension> does not split. If $V=W\oplus U$ for a <submodule> $U$, then $U$ would be a one-dimensional <trivial Lie algebra representation>, since it maps isomorphically to $V/W$. But $\ker H=kv_0$ and $Fv_0=v_1\ne0$, so no such $U$ exists. Thus
$$
\boxed{\begin{gathered}W=\bigoplus_{i\geq1}kv_i,\qquad V/W\cong k_{\mathrm{triv}},\\\text{Submodules: }0,W,V;\qquad V\text{ is not semisimple}.\end{gathered}}
$$
Here <semisimple module> means a <direct sum of modules> that are <simple modules>, equivalently a module in which every <submodule> has a complementary <submodule>. This is the <nonsplit zero-weight Verma extension for sl2>: rescaling to $w_i=i!v_i$ gives $Fw_i=w_{i+1}$ and $Ew_i=i(1-i)w_{i-1}$. Its <infinite-dimensional vector space> is outside the hypotheses of the <Weyl complete reducibility theorem>.
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