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I want to jot down the name so I don't forget.
Literal
Forget-not [so-that-ように] name [object-を] want-to-note-down-in-advance.
ように (so as to) marks purpose — 'so that I don't forget.' 書き留める means 'to jot down / note.' ~ておく (do in preparation) adds the nuance of doing it ahead of time for later benefit. ~たい expresses the speaker's desire. The three layers (purpose + preparation + desire) stack cleanly.