Etymology
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Kraut (n.)
"a German" (especially a German soldier), 1841, but popularized during World War I, from German kraut "cabbage," considered a characteristic national dish. The "cabbage" sense is attested in English from 1855.
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sauerkraut (n.)

"a favorite German dish consisting of cabbage cut fine, pressed, salted, and fermented until sour," 1630s, from German Sauerkraut, literally "sour cabbage," from sauer "sour" (from Proto-Germanic *sura-; see sour (adj.)) + Kraut "vegetable, cabbage," from Old High German krut, from Proto-Germanic *kruthan.

They pickle it [cabbage] up in all high Germany, with salt and barberies, and so keepe it all the yeere, being commonly the first dish you have served in at table, which they call their sawerkrant. [James Hart, "Klinike, or the diet of the diseased," 1633]

In U.S. slang, figurative use for "a German" dates from 1858 (compare kraut). "The effort to substitute liberty-cabbage for sauerkraut, made by professional patriots in 1918, was a complete failure." [Mencken]. French choucroute (19c.) is the German word, but via Alsatian German surkrut but with folk etymology alteration in French based on chou "cabbage" + croûte "crust" (n.).

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