$\DeclareMathOperator\Der{Der}$Consider the polynomial algebra $R[X] = R[x_1,\ldots, x_n]$ over the ring $R = \mathbb{C}[t]$ and let $\Der_{R}(R[X])$ be the Lie algebra of derivations of $R$-algebra $R[X]$.

Assume that for $n$ derivations $D_i= f_{i1}\partial_{x_1}+\dotsb+f_{in}\partial_{x_n}\in\Der_R(R[X])$, $f_{ij}\in R[X]$ we have that the polynomial $$F=\det(f_{ij})_{i, j=1,\ldots, n}\in R\setminus\{0\}.$$

Is it true that there exists an $R$-Lie algebra $\mathcal{C}\subset\operatorname{LND}(R[X])$ such that the set $\mathcal{C}\cap(\bigoplus_{1\leq i\leq n}R[X]\cdot D_i)$ generates $\bigoplus_{1\leq i\leq n}R[X]D_i$ as an $R[X]$-module?