Compositions, methods, and systems to form vaterite with magnesium oxide

A method to form a composition, comprising:

• (i) calcining limestone to form a mixture comprising lime and magnesium oxide, and a gaseous stream comprising carbon dioxide;
• (ii) dissolving the mixture comprising lime and magnesium oxide in a N-containing salt solution to produce an aqueous solution comprising calcium salt and magnesium oxide and a solid comprising magnesium oxide;
• and
• (iii) treating the aqueous solution comprising calcium salt and magnesium oxide and the solid comprising magnesium oxide with the gaseous stream comprising carbon dioxide to form a composition comprising vaterite and magnesium oxide, and the solid comprising magnesium oxide.