Irreducibility of (anti)self-dual reps

[gentsagree]

Imagine we are talking about the group SO(4). The second rank antisymmetric representation is reducible into self-dual and antiself-dual representations. I think a good way to visualise this is by noticing that the projection of $\Lambda^{2}V$ into self and antiself dual subspaces commutes with the action of SO(4).

However, how can I show that those subspaces are themselves irreducible?

Thanks!

 

[fresh_42]

It would be easier to argue if you supply a more detailed framework. In general a representation is irreducible if there are no proper submodules. So you could take a basis vector and show that its orbit is the whole module.