Thus if v and w are symplectic, using L v ω ω = 0 , Cartan's identity, and the fact that ι ι w ω ω is a closed form, ι ι [ v , w ] ω ω = L v ι ι w ω ω = d ( ι ι v ι ι w ω ω ) + ι ι v d ( ι ι w ω ω ) = d ( ι ι v ι ι w ω ω ) = d ( ω ω ( w , v ) ) .