Translation:I always remember you when I read this book.
Couldn't "I always remember you when reading this book" also work? Or do you have to have the second "I" in the sentence?
It's less about the second "I" than about the whole grammatical structure.
"I always remember you while/when reading this book" has a direct analogue in Russian:
"Я всегда вспоминаю тебя, читая эту книгу."
I'll grant you that the meaning of the two sentences (the suggested one & yours) is nearly identical, but Duo is often picky insisting on preserving the grammatical structure whenever possible. It is definitely possible here, so why do you feel compelled to deviate from it?
Bcпоминать/вспомнить (imperfective/perfective) - to retrieve from memory, to recall. "To remember" works in some contexts (like this one), but I would not generically translate "вспоминать" as "to remember".
Помнить - to retain in memory, the most generic meaning of "to remember".
Запоминать/запомнить (imperfective/perfective) - to memorise, to store something in memory.
It should be translated as either I'm always reminded of you or I always think of you. The suggested "I always remember you" implies that you forget about the person altogether, and are reminded of its existence upon reading the book. This is not the most plausible interpretation of the situation.
I'm here just to share that some days ago in the middle of the night I had what was to me a great insight: that Всегда, meaning always, consists of the word "всё" (all, everything), followed by the particle "гда", that shows up in time related words (like когда, никогда, иногда, тогда). I was very happy with that =). Has someone else thought of that?