Where does “steel” come from?

I need to flag an issue with this chain before writing the sentence: the etymology provided doesn't actually support "steel" (the metal alloy). The chain shows "Bastille" → "stēle" (Old English) → "stilus" (Latin, a writing instrument or stake) → "steyg-" (PIE, meaning to pierce or stick). This traces a word meaning "stake/pointed object," not the metal steel. Steel (the iron alloy) has a different etymology entirely, typically traced through Old English *stȳle or similar forms to Proto-Germanic, ultimately related to PIE *stel- (to stand/be firm) or *stek- (to cover/thatch)—not the *steyg- chain provided. **I cannot write an accurate one-sentence summary with this chain, as it does not etymologically support the headword "steel."** Could you verify

steel (English): An artificial metal produced from iron, harder...

Definitions

  1. An artificial metal produced from iron, harder...

Ancestry of “steel”, step by step

steel traces back through two separate lines of ancestry.

via English Bastille

StepLanguageWordMeaning
1EnglishBastilleA former fortress and prison in Paris, France, the storming of which in 1789 began the French Revolution
2FrenchBastilleBastille
3Middle Frenchbastille
4Old Frenchbaastel
5Occitanbastida
6Occitanbastirto build; to construct
7Vulgar Latin*bastīre
8Frankishbastijanto sew, weave; to weave, plait, sew; to plait,...

via Middle English stele

StepLanguageWordMeaning
1Middle Englishstelesteel
2Old Englishstelastalk (of a plant)
3Proto-Germanicstelanąto steal; to sneak
4Proto-Indo-Europeantsel-to sneak

Words derived from “steel

Every word from Frankish bastijanEvery word from Vulgar Latin *bastīreEvery word from Occitan bastir
steel — Etymology, Origin & Word History | Etymology Explorer