Captaincy General of the Philippines administrative units: Spanish rule in the Philippines ceased in 1898 after the war with the United States, which annexed most territories, although the administrative jurisdictions remained intact. Most of the remaining territories in the Pacific Ocean were sold to Germany during the German-Spanish Treaty of 1899. Captaincy General of the Philippines The Captaincy General of the Philippines ( ; ) was an administrative district of the Spanish Empire in Southeast Asia governed by a Governor-General. The Captaincy General encompassed the Spanish East Indies, which included among others the Philippine Islands and the Caroline Islands. It was founded in 1565 with
Lie group Lie groups are called "isomorphic" if there exists a bijective homomorphism between them whose inverse is also a Lie group homomorphism. Equivalently, it is a diffeomorphism which is also a group homomorphism. Isomorphic Lie groups necessarily have isomorphic Lie algebras; it is then reasonable to ask how isomorphism classes of Lie groups relate to isomorphism classes of Lie algebras. The first result in this direction is Lie's third theorem, which states that every finite-dimensional, real Lie algebra is the Lie algebra of some (linear) Lie group. One way to prove Lie's third theorem is to use Ado's theorem, which says