Let $\mathcal G$ be the diffeomorphism group of the unit circle in the complex plane. Since the unit circle is a compact smooth manifold, $\mathcal G$ can be given the structure of a Lie group such that its Lie algebra is the space of smooth vectors fields on the unit circle. Write $\mathcal G$ also for this Lie group.
Let $\mathfrak g$ be the (abstract) Lie algebra over $\mathbb C$ generated by elements $\{L_n\}_{n\in\mathbb Z}$ satisfying $$ [L_n,L_m]=(n-m)L_{n+m}. $$ (This abstract Lie algebra is known as the Witt algebra. One way to realize the Witt algebra is by taking $L_n$ to be the operator $z^{1-n}\frac{d}{dz}$ acting on the Laurent polynomial ring $\mathbb C[z,z^{-1}]$.)
My question is the following: $$ \text{Does the Lie algebra of $\mathcal G$ contain a subalgebra isomorphic to $\mathfrak g$?} $$
According to the discussion in Section 5.4 of Schottenloher's book "A Mathematical Introduction to Conformal Field Theory", the answer would seem to be "no". But according to the survey article "Kac-Moody and Virasoro algebras in relation to quantum physics" by Goddard and Olive, the answer seems to be "yes" (page 14).
I suspect there is some subtle language difference between the mathematicians and the physicists. The question I have asked above is mathematically precise, and I am therefore looking for a mathematically precise answer.