Mexican settlement in the Philippines territory subject to the designs of Washington, D.C. During the Spanish period, the islands formed part of the Viceroyalty of New Spain, coupled with other areas of the Pacific Ocean such as the Marianas and the Caroline Islands and during a short period in northern Taiwan. The Spaniards made trade routes from Mexico to the Philippines, the main ports of Mexico today to start their starting points were Acapulco and Puerto Vallarta with their final destination to Manila, the current capital of the Philippines. The Spanish ships on these routes were known as the Manila galleons. After the arrival of
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