Suppose $\mathfrak{g}$ semisimple Lie algebra and $M$ is a $\mathfrak{g}$-module, a Verma flag (or standard filtration) is a filtration $M=M_0>M_1> \ldots >M_n=0$ such that $M_i/M_{i+1} \simeq \Delta(\mu_i) $ a Verma module, for some weight $\mu_i \in \mathfrak{h}^*$. I have to prove that if $M=N\oplus L$ then both $N,L$ have a Verma flag as well. My attempt is to prove that is $\pi_1$ is the projection to the first component and $N_i=\pi_1(M_i)$, then $N=N_0\ge N_1 \ge \ldots \ge N_0$ is a Verma flag for $N$. I work before with the first two terms, we have $\Delta(\mu) \simeq M/M_1\simeq N/N_1\oplus L/L_1 \xrightarrow{\pi_1}N/N_1$, so there is a surjection of $\Delta(\mu)$ to $N/N_1$. Consider $(v+N_1,w+L_1)$ the $\mathcal{U}(\mathfrak{n^{-}})$-free generator of $N/N_1\oplus L/L_1$, then I have to prove that $N/N_1$ is $\mathcal{U}(\mathfrak{n^{-}})$-free on $v+N_1$. Let a non zero $y\in \mathcal{U}(\mathfrak{n^{-}})$, suppose $y.(v+N_1)=N_1$, then $y.v\in N_1$. From this we deduce that $y.(v+w+M_1)\in (L+M_1)/M_1\simeq L/L_1$, but I can't deduce more information to go further.
We can rephrase the problem in a problem in theory of modules: let $M$ a $\mathcal{U}(\mathfrak{n^{-}})$-module of rank $1$ free on $u\in M$. Suppose $m=N\oplus L$ and write uniquely $u=v+w$ with $v\in N$ and $w\in L$. Then $N$ is again a $\mathcal{U}(\mathfrak{n^{-}})$-module free on $v$.
The proof, done by induction on the length of a Verma flag, can be found in chapter $3.7$ of "Representations of Semisimple Lie Algebras in the BGG Cathegory $\mathcal{O}$", J.E. Humphreys. Fortunately the chapter is even available on Google Books at this link.