Now, the colour of the Lans is a little hard to pinpoint. At times, it feels like it’s meant to be blue, which would be fitting, as the term for blue is, guess what, lán (藍), and at times - like when Wei Wuxian repeatedly calls their attire “mourning attire”, it feels like it’s white (白, bái)
However, it should be noted that I believe that Lan Wangji - and to a lesser extent Sizhui- is the only one explicitly noted as wearing white, so we’ll be treating the colours accordingly - Lan Wangji and Sizhui are white, but the Lans in general are blue
Immediately, this tells us that Wangji and Sizhui are, in some way, different from the Lans around them (Wangji because of his decisions, and Sizhui because he was born a Wen), not in the least because the word for blue is lán, literally pulling them away from the rest of the Lans
But, first, let’s tackle their hues - what is white ? White, or 白, is the symbol for Metal on the WuXing, and is directly opposed to Wood, which is blue (again, another hint at Lan Wangji’s stance running counter to his clan’s). Between the two hues stands Yellow, metaphorically dividing them (much like Jin Guangyao divided the Twin Jades into who liked him (Xichen) and who didn’t (Wangji))
It is a mourning hue, which is fitting - as both Wangji and Sizhui are in mourning at almost all times, even as the nominal association with blue presents them as healed and calm.
It is also the colour of purity and innocence (both Lan Wangji and Sizhui being fully innocent, despite the latter being treated with suspicion due to his birth as a Wen, and the former punished for his association with Wei Wuxian), and having only pure intentions
The element associated with it, Metal (金, ironically jīn) is associated with dusk, with the West, with loss and grief. Its direction is going inwards and its energy is contracting (much like how Lan Wangji secluded himself from the world after Wei Wuxian’s death). Despite this, it’s association with silver and gold also makes it an emblem of valuable things
Attributes of Metal are firmness, rigidity, focus, integrity, and quality (very apt for Wangji). When in excess, it can make people controlling, cutoff and arrogant, set in their ways (perhaps representing how, despite everything, Wangji is a relative of Lan Qiren and a member of the Gusu Lan)
They are self-reliant and resilient and prefer to handle their problems alone (all of that sounds about right). He is also intelligent and good at organisation and stability, and can change and transform those who come into contact with them (again, sounds about right). The well-balanced metal person is calm, humble and honest as well as a good person with a focused mind. (yup)
In the regenerative cycle of WuXing, Earth brings about Metal, in much the same way the Jin sect brought about Lan Wangji and Lan Sizhui’s grief. It also brings about Water (again, remember who wears the black robes associated with water ?)
In the conquest cycle, Fire overcomes Metal (the Wen sect killed Lan Wangji’s sect and set in motion the series of events that caused the death of his beloved).
Metal can also add minerals to Wood (blue/green) to help it grow strong and steady (in much the same way Lan Wangji’s actions do end up bolstering the reputation of his Lan Sect, and how they end up helping Nie Huaisang), but can also topple it (like how Wangji’s revealing of the truth behind Guangyao caused his brother immense grief and caused him to recede from the world)
However, it is said that a lot of wood can defeat small amounts of metal
Wood by excessively wanting to grow undermines its integrity and becomes a weed and not a tree (Su She exemplified) or as an axe would be broken or dulled after trying to chop down a forest. (like, how, despite trying his best, Lan Wangji was unable to stop his sect from attacking Wei Wuxian)
Now, for the actual Sect colour of the Lan Sect, either emerald-blue (碧 bì) or blue-green (青, qīng)
Starting, as usual, with the simpler colour, emerald-blue is a member of the wǔjiànsè, and is a combination of eastern blue-green and western white. It could also be said to represent the Lan sect encompassing both it’s general members (blue) and outliers like Wangji and Sizhui (white)
As for qīng, it is associated most with health, prosperity, and harmony (sounds about right for the Lan). It is used in places like the Temple of Heaven to represent Heaven. It also stands to represent Wood on the WuXing
As for modern blue (蓝 lán ), it isn’t without meaning. Generally, it is associated with calm, health, and longevity, as well as dependability and peace (which is the way most of the world view Xichen). It is also seen as a symbol of trust, credibility and reliability (representative of both the trust the cultivation world had in Xichen and the trust that doomed him)
Metaphorically, I feel like qīng represents the Lan Sect at large, and its aspirations and birth, while bì represents Lan Xichen before the Siege, and lán after.
It represents how, while both Lan Wangji and Lan Xichen were once distinct from the Lan Sect, Lan Xichen leaned much closer to the sect than his brother, standing between the pair as a metaphorical bridge between them. However, after the Siege, that bridge was burned as Lan Xichen and Lan Wangji began leaning opposite ways, and, while Lan Xichen’s reputation grew stronger, it was at the cost of becoming fully “Lan” in a way (though still consciously separate), which is represented by the colour associated with him being literally “lan”