Not dangerous — just less forgiving. The skin around the eyes is the thinnest on the face, so the same retinol that your cheeks shrug off can sting and flake there. But "delicate, go carefully" is very different from "never": the landmark tretinoin photoaging work assessed facial wrinkling including the eye area, and crow's feet are precisely the kind of fine lines retinoids have the best evidence for (https://pubmed.ncbi.nlm.nih.gov/3336176/).
How to do it without a week of red crepe-paper skin: use a rice-grain amount for both eye areas combined, keep it on the orbital bone (the ridge you can feel) rather than up against the lash line — product migrates a little on its own — and start one or two nights a week. Buffering is your friend here: moisturizer first, retinol over it, which softens the hit without meaningfully blunting the effect (https://www.dermatologytimes.com/view/-open-sandwich-moisturization-regimen-does-not-affect-bioactivity-of-retinols-and-retinoids). Fully dry skin only; damp skin increases penetration and sting.
On the expensive retinol eye creams: they're mostly your regular retinol at a lower concentration in a richer base, pre-buffered for you. Convenient, not magic. Doing the same thing with what you own is legitimate.
Expect the same honest timeline as the rest of the face — months, not weeks (https://pubmed.ncbi.nlm.nih.gov/2024983/) — and wear sunscreen daily, since the eye area shows sun damage first. If you get persistent redness or the skin ever feels raw, back off frequency; slow around the eyes is still faster than quitting.