Complex sentence: At a formal, international level the war (1864-1870) was fought between the Republic of Paraguay and the Empire of Brazil, the latter soon being joined in a triple alliance with the Argentine Republic and the Republic of Uruguay.
Split into simpler sentences: At a formal, international level the war was fought, at first (1864), between the Republic of Paraguay and the Empire of Brazil. Soon it was continued (1865-1870) between Paraguay and a triple alliance of Brazil, the Argentine Republic and the Republic of Uruguay.