What are the word classes of the adverbial "[...] which is why [...]"? is cannot be a lexical verb, can it? I thought which is a relative pronoun, and why is an adverb of reason.
The question becomes clearer if a concrete example is given, e.g.
(1) They laughed at me, which is why I left.
Yes, the which here is a relative pronoun. The unique thing about it is that it takes the entire matrix clause as its antecedent -- most relative pronouns have just a noun as (part of) their antecedent. And yes, the wh-word why is an interrogatve adverb; in this case, it functions as a predicative expression, similar to how a predicative adjective or noun functions. Given these considerations, there is nothing particularly challenging about the formulation.
The answer to the question is therefore that the expression ...which is why... is not a single unit, but rather it is compositional, made up of smaller parts that one can identify and combine productively.
Merriam-Webster lists how and why as adverbs. But in several cases, they appear to take a role that mirrors that of the pronoun what, as a noun phrase head introducing a relative clause. Compare these examples that illustrate how how and why fill the noun-like role commonly associated with a pronoun:
- "...which is why we do it"
- "This is why we do it"
- "This is how we do it"
- "This is what we do [t]" (where [t] is a trace)
- "How you're doing it is not how you're supposed to do it"
- "Why you did it doesn't matter"
- "I'm telling you why it's a pronoun"
Here, what means "the thing that", why means "the reason that", and how means "the way that".