Let M be a symplectic manifold of dimension $2n$ and $TM$ denote its tangent bundle. Let Sp(TM) denote the bundle over M whose fibers are linear maps preserving symplectic structure on M. Is Sp(TM) trivial (i.e can it be written as $M \times Sp(2n)$) because Id gives a cross section and existence of a global cross section imply the fiber bundle is trivial?
Edit: I will include more context. I want to understand the paper https://arxiv.org/pdf/1305.6810.pdf , what I am asking is defined at page 5 around the middle of the page starting with Letting Sp(TM).. Now I am thinking it is perhaps not a principal G bundle it is maybe rather just a fiber bundle with a fiber equal to Lie Group and since there is no equivariant G action, I cannot use the existence of Id section to trivialize the bundle. The concepts are rather new to me, so it would be great if someone can present their perspective.
Adding this in response to Jason DeVito's comment: $\operatorname{Sp}(TM)$ as defined is not a principal fibre bundle. In fact, it is an associated bundle to the symplectic frame bundle (which is principal).
To see this: let me use the notation $\mathcal{F}_{\operatorname{Sp}}(M)$ for the symplectic frame bundle of $M$. One way of defining $\mathcal{F}_{\operatorname{Sp}}(M)$ is that its fibre $\mathcal{F}_{\operatorname{Sp}}(M)_p$ over $p\in M$ consists of all symplectic linear isomorphisms from $\mathbb{R}^{2n}$ with its standard symplectic form $\Omega$ to $(T_pM,\omega_p)$
$$ \mathcal{F}_{\operatorname{Sp}}(M)_p = \{ b_p:\mathbb{R}^{2n}\to T_pM \mid b_p^*\omega_p = \Omega\} $$
(see for example Metaplectic-c Quantomorphisms by Jennifer Vaughan, Section 3.2, where the symplectic frame bundle is denoted $\operatorname{Sp}(M,\omega)$). This is a principal $\operatorname{Sp}(2n,\mathbb{R})$-bundle, with action given by composition
$$ (b_p, T)\in \mathcal{F}_{\operatorname{Sp}}(M) \times \operatorname{Sp}(2n,\mathbb{R}) \mapsto b_p\circ T \in \mathcal{F}_{\operatorname{Sp}}(M). $$
I claim that (your) $\operatorname{Sp}(TM)$ is isomorphic to the associated bundle $$ \mathcal{F}_{\operatorname{Sp}}(M)\times_{\operatorname{Sp}(2n,\mathbb{R})} \operatorname{Sp}(2n,\mathbb{R}) $$ where the action of $\operatorname{Sp}(2n,\mathbb{R})$ on itself is via conjugation. That is, for $b_p\in \mathcal{F}_{\operatorname{Sp}}(M), S\in\operatorname{Sp}(2n,\mathbb{R})$, the associated bundle is the quotient under the equivalence relation $$ (b_p, S) \sim (b_p\circ T, T^{-1}\circ S \circ T) $$ with $T\in \operatorname{Sp}(2n,\mathbb{R})$. The isomorphism between this associated bundle and $\operatorname{Sp}(TM)$ is given by $$ [b_p,S] \mapsto b_p\circ S\circ b_p^{-1} $$ where $[b_p,S]$ denotes the equivalence class of $(b_p,S)$. It is not too difficult to check that this is well-defined, and a bijection.
This hopefully clarifies that $\operatorname{Sp}(TM)$ is not a principal bundle, but rather an associated fibre bundle.