I am currently studying the paper
Parker, Thomas; Taubes, Clifford Henry, On Witten’s proof of the positive energy theorem, Commun. Math. Phys. 84, 223-238 (1982). ZBL0528.58040. and have a question concerning the beginning of section 2:
Denote by $Cl_{3,1}$ the CLifford algebra of $\mathbb{R}^{1,3}$ with respect to a metric of signature (3,1). We consider the spinor group Spin$_{3,1}$, which is isomorphic to the special linear group SL$(2,\mathbb{C})$. The (non-conjugate) spinor representation is then given by an irreducible representation of the even subalgebra $Cl^0_{3,1}$ over $\mathbb{C}$ $$ Cl^0_{3,1} \cong End_{\mathbb{C}}(V)$$ such that the volume element $\omega = e_0...e_3$ acts as $i$ on $V$ ($e_0,...,e_3$ denoting the standard basis of $\mathbb{R}^{1,3}$).
Now, it is claimed that the representation $V$ carries an invariant symplectic form $\sigma$. This is equivalent to the existence of an inner product $$(-,-): V \times V \rightarrow \mathbb{C},$$ expressed in matrix form by an invertible skew-hermitian matrix.
I am very puzzled by this as I am failing to see how this is induced by the described representation. Any help would be greatly appreciated.