Split Lie Algebra
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In the mathematical field of Lie theory, a split Lie algebra is a pair (mathfrak{g}, mathfrak{h}) where mathfrak{g} is a Lie algebra and mathfrak{h} mathfrak{g} is a splitting Cartan subalgebra, where "splitting" means that for all x in mathfrak{h}, operatorname{ad}_{mathfrak{g}} h is triangularizable. If a Lie algebra admits a splitting, it is called a splittable Lie algebra. Note that for reductive Lie algebras, the Cartan subalgebra is required to contain the center. Over an algebraically closed field such as the complex numbers, all semisimple Lie algebras are splittable (indeed, the Carta...
In the mathematical field of Lie theory, a split Lie algebra is a pair (mathfrak{g}, mathfrak{h}) where mathfrak{g} is a Lie algebra and mathfrak{h} mathfrak{g} is a splitting Cartan subalgebra, where "splitting" means that for all x in mathfrak{h}, operatorname{ad}_{mathfrak{g}} h is triangularizable. If a Lie algebra admits a splitting, it is called a splittable Lie algebra. Note that for reductive Lie algebras, the Cartan subalgebra is required to contain the center. Over an algebraically closed field such as the complex numbers, all semisimple Lie algebras are splittable (indeed, the Cartan subalgebra acts not only by triangularizable matrices but a fortiori by diagonalizable ones) and all splittings are conjugate; thus split Lie algebras are of most interest for non-algebraically closed fields.
