n matrices over a field k, for a natural number n. In particular, the multiplicative group Gm is the group GL(1), and so its group Gm(k) of k-rational points is the group k* of nonzero elements of k under multiplication. Another reductive group is the special linear group SL(n) over a field k, the subgroup of matrices with determinant 1. In fact, SL(n) is a simple algebraic group for n at least 2. O(n), SO(n), and Sp(n) An important simple group is the symplectic group Sp(2n) over a field k, the subgroup of GL(2n) that preserves a nondegenerate alternating bilinear form on the vector space k2n. Likewise, the orthogonal group O(q) is the subgroup of the general linear group that preserves a nondegenerate quadratic form q on a vector space over a field k. The algebraic group O(q) has two connected components, and its identity component SO(q) is reductive, in fact simple for q of dimension n at least 3. (For k of characteristic 2 and n odd, the group scheme O(q) is in fact connected but not smooth over k. The simple group SO(q) can always be defined as the maximal smooth connected subgroup of O(q) over k.) When k is algebraically closed, any two (nondegenerate) quadratic forms of the same dimension are isomorphic, and so it is reasonable to call this group SO(n). For a general field k, different quadratic forms of dimension n can yield non-isomorphic simple groups SO(q) over k, although they all have the same base change to the algebraic closure . Tori The group and products of it are called the algebraic tori. They are examples of reductive groups since they embed in through the diagonal, and from this representation, their unipotent radical is trivial. For example, embeds in from the map Non-examples Any unipotent group is not reductive since its unipotent radical is itself. This includes the additive group . The Borel group of has a non-trivial unipotent radical of upper-triangular matrices with on the diagonal. This is an example of a non-reductive group which is not unipotent. Associated reductive group Note that the normality of the unipotent radical implies that the quotient group is reductive. For example, Other characterizations of reductive groups Every compact connected Lie group has a complexification, which is a complex reductive algebraic group. In fact, this construction gives a one-to-one correspondence between compact connected Lie groups and complex