References on the ''bi-symplectic'' structure of $T^\ast T \mathcal M$.

Question

Let $\mathcal M$ be a smooth manifold. One can consider two tautological one forms on $T^\ast (T \mathcal M)$:

1. ("Diagonal case") The usual tautological one form on $T^\ast \mathcal E$, where $\mathcal E = T \mathcal M$ is a generic smooth manifold. The latter can be by definition derived from the trivial duality pairing: \begin{equation*} (p,v) \in T^\ast (T \mathcal M) \times T (T \mathcal M) \mapsto \langle p,v \rangle \end{equation*}
2. ("Anti-Diagonal case") The one form derived from the pairing

\begin{equation*} (p,v) \in T^\ast (T \mathcal M) \times T (T \mathcal M) \mapsto \langle p \quad [\mathop{Im}(d \pi^\ast)] , d \pi v \rangle_{T^\ast \mathcal M , T \mathcal M} \end{equation*}

where $d\pi: TT \mathcal M \to T\mathcal M$ is the differential of the trivial projection $\pi:T \mathcal M \to \mathcal M$; that is in a coordinate chart $U$ of $\mathcal M$ where $q=(x,e)\in T \mathcal M$ and $(p,v)=(p_x,p_e,v_x,v_e) \in T^\ast (T \mathcal M) \times T (T \mathcal M) $:

\begin{equation*} \langle p \quad [\mathop{Im}(d \pi^\ast)] , d \pi v \rangle_{T^\ast \mathcal M , T \mathcal M} = (p_e)_{i} (v_x)^{i}. \end{equation*}

(Soft) Questions:

1. I am struggling to find basic references especially on 2. above. In particular, I wish I could use a good one to avoid redoing all the calculations/checks by myself.

2. Any insight or comment pointing towards a related topic or an associated general structure is welcome.