Etymology
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Words related to brachio-

*mregh-u- 
Proto-Indo-European root meaning "short."

It forms all or part of: abbreviate; abbreviation; abridge; amphibrach; brace; bracelet; brachio-; brachiopod; brachiosaurus; brachy-; brassiere; breviary; brevity; brief; brumal; brume; embrace; merry; mirth; pretzel; vambrace.

It is the hypothetical source of/evidence for its existence is provided by: Greek brakhys "short;" Latin brevis "short, low, little, shallow;" Old Church Slavonic bruzeja "shallow places, shoals;" Gothic gamaurgjan "to shorten."
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brace (n.)
early 14c., "piece of armor for the arms," also "thong, strap for fastening," from Old French brace "arms," also "length measured by two arms" (12c., Modern French bras "arm, power;" brasse "fathom, armful, breaststroke"), from Latin bracchia, plural of bracchium "an arm, a forearm," from Greek brakhion "an arm" (see brachio-).

Meaning "that which holds two or more things firmly together" (on notion of clasping arms) is from mid-15c. Hence applied to various devices for fastening and tightening. Meaning "a prop, support," especially in architecture, is from 1520s. Of dogs, ducks, pistols, etc., "a couple, a pair" from c. 1400. Braces is from 1798 as "straps passing over the shoulders to hold up the trousers;" from 1945 as "wires for straightening the teeth."
bracelet (n.)
"ornamental ring or clasped chain for the wrist," mid-15c., from Old French bracelet (14c.), diminutive of bracel, from Latin bracchiale "armlet," from bracchium "an arm, a forearm," from Greek brakhion "an arm" (see brachio-).
brachial (adj.)
"belonging to the arm, fore-leg, wing," etc., 1570s, from Latin brachialis, from brachium (see brachio-).
brachiopod (n.)
type of bivalve mollusk of the class Brachiopoda, 1836, Modern Latin, from Greek brakhion "an arm" (see brachio-) + pous "foot" (from PIE root *ped- "foot"). They develop long spiral "arms" from either side of their mouths.
brachiosaurus (n.)
1903, Modern Latin, from Greek brakhion "an arm" (see brachio-) + -saurus. The forelegs are notably longer than the hind legs.
brassiere (n.)
"form-fitting undergarment to support a woman's breasts," by 1902, a euphemistic borrowing in the garment trade, from French brassière "child's chemise; shoulder strap" (17c.), from Old French braciere "arm guard" (14c.), from bras "an arm," from Latin bracchium "an arm," from Greek brakhion "an arm" (see brachio-). The French word was used 18c. in the sense "woman's underbodice."
embrace (v.)
mid-14c., "clasp in the arms," from Old French embracier (12c., Modern French embrasser) "clasp in the arms, enclose; covet, handle, cope with," from assimilated form of en- "in" (see en- (1)) + brace, braz "the arms," from Latin bracchium (neuter plural brachia) "an arm, a forearm," from Greek brakhion "an arm" (see brachio-). Related: Embraced; embracing; embraceable. Replaced Old English clyppan (see clip (v.2)), also fæðm (see fathom (v.)). Sexual sense is from 1590s.
pretzel (n.)

1836, "small, crisp biscuit in the form of a knot, salted on the outside," from German Prezel, also Brezel, from Middle High German brezel, prezel, from Old High German brezitella, brecedela, from Medieval Latin *brachitella, presumably a kind of biscuit baked in the shape of folded arms (source also of Italian bracciatella, Old Provençal brassadel), diminutive of Latin bracchiatus "with branches, with arms," from bracchium "an arm, a forearm," from Greek brakhion "an arm" (see brachio-).

The figurative sense of "a thing much twisted" is by 1945.  Soft pretzels are attested by 1893 in the German regions of Pennsylvania, but did not become widely popular until mid-20c.