Dual anisymmetric tensor generates su(2)

Question

An anisymmetric tensor in so(4) can be split in a self-dual and an anti-dual part. The self dual tensors are the ones that satisfy: $$J_{ab}=\frac{1}{2}\epsilon_{abcd}J_{cd}$$ While the anti-dual satisfies: $$J_{ab}=-\frac{1}{2}\epsilon_{abcd}J_{cd}$$ I've proven that they conmute with the action of so(4). Now the author claims that they each generate an su(2) so that $$so(4)=su(2)\oplus su(2)=so(3)\oplus so(3)$$ I'm trying to figure out how does he come to this conclusion. How can I get an isomorphism between each part and su(2)?