Here we consider the case of unitary highest weight representations: there is a primary state $\left|\phi\right>$ and the representation $V$ is spanned by $L_{-k_1}\dots L_{-k_n}\left|\phi\right>$. Then the quasi-primary states are those annilated by $L_1$ and the derivative is $L_{-1}$.

Let $W=\ker(L_1)$ our claim is $V=W\oplus L_{-1}W\oplus L_{-1}^2W\oplus \dots$. To prove this we make use of grading: $L_{-k_1}\dots L_{-k_n}\left|\phi\right>$ has grade $k_1+\dots+k_n$. So $L_1$ decrease grading by $1$ and $L_{-1}$ increase grading by $1$.

So look at the $n$ part $V_n$, and treat $L_1$ be an operator from $V_n$ to $V_{n-1}$, and $L_{-1}$ be an operator from $V_{n-1}$ to $V_{n}$. with induction we only need to prove $V_n=\ker(L_1)\oplus\operatorname{Img}(L_{-1})$. Which follows from $L_{-1}=L_1^{\dagger}$.