More properties of the Fibonacci word on an infinite alphabet

Recently the Fibonacci word $W$ on an infinite alphabet was introduced by [Zhang et al., Electronic J. Combinatorics 24-2 (2017) #P2.52] as a fixed point of the morphism $\phi: (2i) \mapsto (2i)(2i+ 1),\ (2i+ 1) \mapsto (2i+ 2)$ over all $i \in \mathbb{N}$. In this paper we investigate the occurrence of squares, palindromes, and Lyndon factors in this infinite word.