Can every real Lie algebra be viewed as a Lie algebra of vector fields defined on some manifold?

Question

I know that for any finite-dimensional Lie algebra $L$ over real there is some matrix Lie group $G$ whose Lie algebra is the given $L$ (Ado theorem), that is, it can be interpreted in the Lie algebra of left invariant vector fields of $G$. More speaking, given any (infinite-dimensional) real Lie algebra $L$ can we find a smooth manifold $M$ with $\mathfrak{X}(M)=L$, where $\mathfrak{X}(M)$ is the Lie algebra of vector fields of $M$? If not, what is an assumption for $L$ so that it holds?