In chapter 2 of the paper Symmetry of massive Rarita-Schwinger fields by T. Pilling mentions "the usual" spin projection operators. However, to me, they are not usual and I struggle with intution and notation.

I understand that we find the correct Lorentz representation of the RS vector-spinor by taking the tensor product of a bispinor and vector representation (eq 1 in the paper):

$$\left[(1/2,0)\oplus(0,1/2)\right] \otimes (1/2, 1/2) = (1, 1/2) \oplus (1/2, 1) \oplus (1/2, 0) \oplus (0, 1/2) \quad\quad\quad(1)$$

My first confusion comes from the spin decomposition they use, i.e.

$$(1+0) \otimes 1/2 = 3/2 + 1/2 + 1/2$$

Why are they using the reducible representation $(1+0)$ instead of just $1$? They then talk about two spin-1/2 components. Now, are they referring to the two components that come by constructing it from the reducible rep? Or are they talking about the two spinor (as in Weyl spinor) reps in the Lorentz rep? If I take the two Weyl spinors to be a Dirac bispinor, and I do the spin decomposition starting from the irreducible rep, i.e. $1 \otimes 1/2 = 3/2 + 1/2$, then there's only one spin-component. However, if they want to label this into two Weyl components, that's fine for me.

My main question is about the later mentioned projection operators. Clearly a RS-spinor $\psi_\mu$ has a mixture of 3/2 and 1/2 degrees of freedom. We are not interested in the 1/2 background, so we need a projection operator $P^{3/2}$ to get rid of them. Fine. Likewise, I can define an operator $P^{1/2}$ to get the spin-1/2 background. That's just a mathematical excercise. Now, what are the extra indices at $P^{1/2}_{11}$, $P^{1/2}_{12}$, $P^{1/2}_{21}$ and $P^{1/2}_{22}$? In the paper they are described as

the individual projection operators for the two different spin-1/2 components of the Rarita-Schwinger field

This is where I'm lost. What am I projecting? For what reason? Can someone explain this to me and give me some intution?