Rarita-Schwinger spin projection operators

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 intuition 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):

$$left[(1/2,0)oplus(0,1/2)right] otimes (1/2, 1/2) quadquadquad$$
$$quadquadquad = (1, 1/2) oplus (1/2, 1) oplus (1/2, 0) oplus (0, 1/2) tag{1}$$

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?

edit:

To clarify, consider the Projector

$$P_x = begin{pmatrix}1 & 0 & 0 0 & 0 & 0 0 & 0 & 0end{pmatrix}$$

The intution here is, that is projects the x-component of a three vector. Using this analogy, what is $P^{1/2}_{11}$ projecting?