This is really a comment try to make clear the point Tilman was trying to make but it is too long. A K3 surface has trivial canonical bundle (after all that and simple connectivity is the definition) and hence the bundle of self dual two forms is trivial (since on a complex surface we have $\Lambda^+= \Lambda^{2,0} \oplus R \omega$, $\omega$ being the Kahler form). In fact is follows from Yau's theorem that K3 surfaces admit hyperkahler metrics so there is a metric where the Levi-Civita connection is trivial on $\Lambda^+$.
Two forms act on vector fields on a four-manifold via contraction then duality under this actions self-dual forms act like imaginary quaternions (so quaternions do figure in the story). Thus taking a orthonormal basis of self-dual forms $\omega_1,\omega_2,\omega_3$ and your vector field $X$ you get a framing (not stable) away from the zeros of $X$, by looking at $(X,(\iota_X \omega_1)^*,(\iota_X \omega_2)^*,(\iota_X \omega_3)^*)$. Then arrange that the vector field is pointing out around each little 3-sphere surrounding a zero then you see that the induced framing of each little sphere is the Lie-group framing. I believe this observation is due to Atiyah.