How do you write the prefix "on"?
The prefix "after" with introductory words or adverbs is always written through a hyphen (sisterly, in your opinion, apparently, etc.). In all other cases this prefix is written together.
How to write "by": individual cases
The availability of an adverb (including pronominal) withthe prefix "on" is not the only rule, according to which a complex word is written through a hyphen. The adverbs are written in the same way (with a hyphen), which end in: -y, -oh, -ki, -em (bearish, human, dog-like, Georgian). When this condition is absent, the adverb from "to" is written together: (gradually, slowly).
However, it is worth to distinguish, as written "by" with adverbs or cases, when "on" is a pretext.
- Did (how?) In a new way (adverb).
- Went on (to what?) A new (adjective) way.
In this example, "by" is an excuse, not a prefix. As a preposition "to" with all the words is spelled separately.