What makes Lich Bane a situational item?

because her passive wants her to AA in between abilities. so Lich bane goes with that style. but most people just go for the burst kill.

split pushing helps with towers and your passive dose more damage

Some games you're more comfortable getting close enough to proc the spellblade effect. Mid-late game Lux never REALLY needs to worry about getting in range for Illumination procs, so if you're facing a team you don't need to get close to you can skip the Bane, but sometimes you want to pump up your damage and feel comfortable taking that risk his game, which is a situational thing.

Is the situation one where you frequently find yourself getting into AA range as part of your combo? Then perhaps consider lich bane.

I don't think it's a strong choice most of the time, since while early game weaving in AA's as part of her burst leaves a dent due to her passive, later game she's prone to being able to do a ton of damage from well outside of AA range, and moving into AA range mid-late game is probably rather dangerous for one of the squishier mages in the game.

Most ap champs don't really weave spells, and theres some who buy it because they have some form of auto attack modifier/skill like for example

or just some really spammy ones like

its not a necessity because it has less flat ap then any of the 100 AP items (those are usually chosen based on preference for lux) But if she gets ahead in lane and doesn't need defensive items such as or then it is better to go for the lich bane in that case