Baelily and Mattily are two informal, non-canon, out-of-character-only portmanteau nicknames for two different prospective couples involving Lily Brightendale, both spawning during Athalia Academy Season One. The first, Baelily, is a portmanteau of "Bael" and "Lily", representing the potential romantic couple of Bael and Lily. The second, Mattily, is a portmanteau of "Matt" and "Lily", representing the potential romantic couple of Matthew Terrison and Lily.
Both of these portmanteau nicknames are permanently linked to one another, as they were coined at the same time and always used competitively, due mainly to the fact that Matt and Bael were indirectly competing for Lily's affection in the coinciding in-character events of the time. Most of the Athalia Academy players involved in Season One sided with "Team Baelily" or "Team Mattily", often expressing their allegiance in "I ship Baelily/Mattily" format.
During Season One it was uncertain which camp HylianRose, Lily Brightendale's creator and player, would side with, and the majority of players sided with Team Baelily. There was never a clear conclusion which of the two HylianRose favored, because they dissolved and the concept of Carlily came into play before any progress was made.
While Matt had known Lily ever since Kamasyia Academy (two seasons before Kalmaysia, and three before Athalia Season One), Bael only first met Lily at the beginning of Athalia Season One. However, since Lily was regressed to a younger version of herself by the demon seal, she lost all memory of Matt, giving Bael a chance. Apparently, it was a very large chance, as Matt spent his time trying to transform Lily into her correct form instead of trying to make her love him like Bael did.
At the end of Season One, Bael (a demon) told Lily that Maria Cross could turn her back by reversing the seal, then returned to Hell with the rest of the demon army. The name of Season Two (Finding Bael) derived from the season's central plot involving getting Bael back.
Baelily was the first of the pair to lose traction during Season Two, as Bael's player, Nero, was nowhere to be found for the duration of the season. Mattily remained strong for a while, but when Matt missed several opportunities to advance his cause, his momentum faltered. The idea of Carlily was introduced at about the same time, and this further hampered the phoenix's quest. By Season Three, both Baelily and Mattily were all but dead as Matt's player, Ebil Bunny, also went missing, and Carlily had gained a lot of support. Soon after Season Four, Carlily was officially established in canon, cementing the fate of all other Lily pairings.