Albeit means "although", and it goes before a short describing word or phrase, not a full sentence: "promising, albeit preliminary" means "promising, although it is preliminary". It is formal and mostly written — reach for it (over plain "although") when you want to slip a brief concession onto the same sentence without opening a whole new clause.
albeit: a formal "although", before a short phrase
The results are promising, albeit preliminary.
She accepted the role, albeit with some hesitation.
He agreed, albeit reluctantly, to the new terms.
Not "The results are promising, albeit they are preliminary. (old-fashioned, and widely criticized)" — say "The results are promising, albeit preliminary.". Put albeit in front of a short phrase (albeit preliminary). A full sentence after it does turn up in older formal writing, but it reads as dated — use although there instead.
Try it: The new design is an improvement, ___ a costly one. (albeit / despite / nevertheless) Answer: albeit — Albeit means although and fits right before a short phrase: albeit a costly one. "Despite" needs a noun and would not link a short add-on phrase this way. "Nevertheless" starts a new statement, but here you are tucking a small qualification onto the same one.
notwithstanding: a formal "despite", before a thing
Notwithstanding means "despite" and goes before a noun phrase, not a full sentence. "Notwithstanding the risks" means "despite the risks". It can open the sentence or follow the thing it refers to. It is very formal — reach for it over "despite" in academic, legal or official writing; in everyday speech "despite" or "in spite of" sounds more natural.
Notwithstanding the high costs, we declined the offer.
Notwithstanding her objections, the plan went ahead.
The team pressed on, the setbacks notwithstanding.
Not "Notwithstanding the costs were high, we declined." — say "Notwithstanding the high costs, we declined.". Notwithstanding takes a noun phrase (the high costs), not a full sentence ("the costs were high") — for a full sentence use "although".
Try it: ___ the delays, the committee delivered its report on schedule. (Notwithstanding / Albeit / Lest) Answer: Notwithstanding — Notwithstanding means despite and goes before a noun phrase: Notwithstanding the delays. "Albeit" means although and goes before a short describing phrase, not a plain thing like the delays. "Lest" means for fear that or in case, a different idea altogether.
Try it: albeit links a short describing word or phrase to the same sentence: "promising, albeit preliminary". notwithstanding goes before a noun phrase and means "despite": "notwithstanding the delays". The proposal passed, ___ the objections raised in committee. (notwithstanding / albeit) Answer: notwithstanding — The blank is followed by a noun phrase (the objections raised in committee), so the despite-word notwithstanding fits. Albeit would need a short describing phrase (albeit a narrow vote), not a plain noun group like this.
lest: a formal "in case", with the base verb
Lest means "for fear that" or "in case (something bad happens)". It is formal and literary — you meet it in careful writing, not casual speech, where "in case" or "so that ... don't" is more natural. After lest, use the base verb (be, forget, wake): "We wrote it down lest we forget", not "lest we forgets" or "lest we will forget". You will also meet lest + should — "She spoke softly lest she should wake the baby" — which is standard, and especially common in British writing. Note that lest always points at something you want to avoid, so it is not a general swap for in case: "Take a book in case you get bored" does not work with lest.
Keep a copy lest the original be lost.
She spoke softly, lest she wake the baby.
We documented everything lest there be a dispute later.
Not "We wrote it down lest we will forget." — say "We wrote it down lest we forget.". After lest, use the base verb (forget) — not "will forget" or "forgets". Lest carries the warning by itself.
Try it: Lest means "for fear that" or "in case", and it takes the base verb. Which one fits? (We documented everything lest there be a dispute later. / We documented everything lest there is a dispute later. / We documented everything lest there will be a dispute later.) Answer: We documented everything lest there be a dispute later. — Lest means for fear that or in case and takes the base verb: lest there be. "Lest there is" uses the everyday form, but lest keeps the plain be. "Lest there will be" adds will, which this formal pattern does not use.