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A green-colored substance formed during electrolysis, wherein the electrolyte was magnesium sulfate and minerals in tap water.

I did not have any leads to stick into the water, so I just put a 9 volt Duracel battery in the solution. The battery didn't leak.

White flakes formed after 1 minute before green substance at 10 minutes. So, what is the green stuff and what are the white flakes?

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The green material is probably an iron(II) salt, corroded from what I assume is the steel composing the positive pole of the battery. If it has a gelatinous appearance, and slowly turned orange-ish over time, my guess would be it's $\ce{Fe(OH)2}$, with any orange color deriving from slow oxidation to various iron(III) species such as the oxide-hydroxides.

Per the above link, the white flakes are possibly that same iron(II) hydroxide, prior to reaction with $\ce{O2}$ to form the green color. Alternatively, it could be $\ce{Mg(OH)2}$, or a related magnesium compound, formed at the cathode (negative terminal) from the hydroxide ion produced by reductive water electrolysis:

$$ \ce{2 H2O + 2 e- -> H2 + 2 OH-} $$

A photograph would be tremendously helpful in facilitating identification.

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