I thought one did not use the subjunctive if it is je and je

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It isn't the subject (je) which determines whether the subjunctive is used or not.

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I have the uneasy feeling that I didn't understand your question properly. Can you repost it in some more detail?

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Tessa, I think I know what you're referring to, and it's a different circumstance.

"When the subject of the subordinate clause introduced by the conjunction que is the same as that of the main clause, and the tense is the same, the infinitive must be used instead of the subjunctive." (Advanced French Grammar, L'Huillier) So "j'ai besoin de manger quelque chose" or "je tiens à partir à six heures." (L'Huillier)

The same would apply to tu, elle, nous, etc., so "tu as besoin de manger...", "elle a besoin de manger...", "nous avons besoin de manger...", and so on.

EDIT: But, "je tiens que tu partes à six heures." The subjects in the main clause and subordinate clause are different.