Let $C^{2}$ be an irreducible representation of the Lie group $SU(2)$. Let us consider the tensor product of three copies of the representation. Then $C^{2}\otimes C^{2}\otimes C^{2}=2C^2\bigoplus C^4$. Now let $H$, hermitian, be invariant under the action of the Lie group generated by the tensor product, i.e., it is invariant under the action of all generators of the group.
I know that I can choose a basis in which the Casimir operator of the group is diagonal and where $H$ is block diagonal. Then I can write $H$ as operators and projectors that respect the subspaces. My question concerns the form of this block-diagonal expansion for the irreducible representation of multiplicity 2. Can I write $H=A_{2}^1P_{2}^1+A_{2}^2P_{2}^2+A_{4}P_{4}$ or is it in general $H=A_{2}P_{2}+A_{4}P_{4}$. In the first form, $P_2^1$ and $P_2^2$ are different projectors to the two subspaces with $C^2$ respectively, and $A_{2}^1$ and $A_{2}^2$ act in the correspondent subspace. $A_{4}\in \mathbb{R}$ and $P_{4}$ is a projection to $C^4$. In the second equation, $A_{2}P_{2}$ mixes the subspace with multiplicity two. If the first equation is true, the second is as well, but I want to be sure if the first equation is the most general result.