complex semisimple Lie algebra admits a unique (up to isomorphism) compact real form . The real forms of a given complex semisimple Lie algebra are frequently labeled by the positive index of inertia of their Killing form.

For example, the complex special linear algebra has two real forms, the real special linear algebra, denoted , and the special unitary algebra, denoted . The first one is noncompact, the so-called split real form, and