condensed matter - Detailed derivation and explanation of the AKLT Hamiltonian

I am trying to read the original paper for the AKLT model,

Rigorous results on valence-bond ground states in antiferromagnets. I Affleck, T Kennedy, RH Lieb and H Tasaki. Phys. Rev. Lett. 59, 799 (1987).

However I am stuck at Eq. $(1)$:

we choose our Hamiltonian to be a sum of projection operators onto spin 2 for each neighboring pair:

$$\begin{align} H &= \sum_i P_2 (\mathbf{S_i} + \mathbf{S_{i+1}}) \\ &= \sum_i \left[\frac{1}{2}\mathbf{S_i}\cdot\mathbf{S_{i+1}} + \frac{1}{6}(\mathbf{S_i}\cdot\mathbf{S_{i+1}})^2 + \frac{1}{3}\right] \end{align}\tag{1}$$

In the equation, $H$ is the proposed Hamiltonian for which the authors intend to show that the ground state is the VBS ground state: the (unique) state with a single valence bond connecting each nearest-neighbor pair of spins, i.e. a type of spin-$1$ valence-bond-solid. Moreover, $\mathbf{S_i}$ and $\mathbf{S_{i+1}}$ are spin-$1$ operators, and $P_2$ is the projection operator onto spin-2 for the pair $(i,i+1)$.

I have several questions here.

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  First, how did the authors come up with the first line by observing the spin-1 valence-bond-solid state as below (Fig. 2 of the above paper)?

  
  

  
  

  

  
  

  
  


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  Why do they use a Hamiltonian which is "a sum of projection operators onto spin 2 for each neighboring pair"?

  
  


• What does it mean exactly to project spin-$1$ pairs to spin $2$, and why do they want to project to spin $2$?

I feel there are quite a few steps skipped between here and standard QM textbooks. It would be great if somebody could recommend me some materials bridging them.

• Secondly, I want to know how to go from the first line to the second line of equation $(1)$. However, I couldn't find the explicit form of $P_2$ either in the paper or by googling. Could somebody give me a hint?

Edit: I found a chapter of the unfinished book "Modern Statistical Mechanics" by Paul Fendley almost directly answers all my questions.