To me, "to seriously take" sounds strange not because it's split infinitive (there's nothing wrong with that), but "seriously" is not just an ordinary adverb here: it's a complement of the verb _take_. "To take the work seriously" is a very different thing from "to take the work," while "to engage the work seriously" is basically the same as "to engage the work," except more seriously.

And I don't think "seriously" (or a similar adverb), when used as a complement of "take," could just move around freely. Just like you can't say "I away chased the dog."

* And, while we're at it, it's a bad idea to take seriously (haha) anyone who thinks split infinitives are wrong. You can find tons of amusing writings by actual linguists here: https://languagelog.ldc.upenn.edu/nll/index.php?s=Split+infi...

(Spoiler alert: they think the rule is BS.)