To divide or to split; also a synonym for adhere.
The verb cleave comes from the Old English cleofan, meaning to spit or to separate and earlier from the Proto Indo European root of gleubh, meaning to cut or to slice.
Modern English cleave also means to adhere, stemming from the Old English clifian. Both senses are still valid nowadays, bit due to their symmetry and following thus- the easy confusion of these two, today these words are mainly replaced with stick and split.
Why this word?
For two main reasons; we slice and we cut things all the time, but we also adhere to things- all the time. There is no reason not to have another word for it. Also, the word cleavage stems from this very root and it is always nicer to use a cleavage related word!
Funny enough, the word cleavage is a relatively new word, coined in 1946 by the “Time” magazine August 5th edition reading “Johnston Office trade term for the shadowed depression dividing an actress’ bosom into two distinct sections”.
How to use cleave?
In the exact way you use split or adhere.
“David is cleaving the onions while I’m chopping the garlic cloves” or “David had a huge fight with the boss after his refusal to cleave to the latest business model“.