Let  $M$ be  a Riemannian  manifold. We denote by $\mathfrak{g}$ the space of  all smooth function $f:TM\to \mathbb{R}$  with fibre wise polynomial growth. Is it a Lie  algebra? What is a precise infinite dimensional Lie group whose  Lie  algebra is the above $\mathfrak{g}$?
Is the isomorphism class of Lie  algebra  mentioned above independent of choosing Riemanian metric?