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Capitalize if part of the proper name:
            Atacama Desert
            Kalahari Desert
            Sonoran Desert
            Western Desert

Do not capitalize if not part of the proper name.

Sahara, Gobi, Negev should be written without desert because these names themselves mean desert.  Write:
            the Sahara not the Sahara Desert; Sahara means "desert" in Arabic.
            the Gobi not Gobi Desert; Gobi means "desert" in Mongolian.
            the Negev not Negev Desert; Negev means "desert" in Hebrew.

although if necessary for clarity desert may be used lowercased:
        Sahara desert
        Gobi desert
        Negev desert