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 $\varphi :(2i)\mapsto (2i)(2i+1),(2i+1)\mapsto (2i+2)$ over all $i\in \mathbb{N}$. In this paper we investigate the occurrence of squares, palindromes, and Lyndon factors in this infinite word.