Split Lie algebra

• Over an algebraically closed field, all Cartan subalgebras are conjugate. Over a non-algebraically closed field, not all Cartan subalgebras are conjugate in general; however, in a splittable semisimple Lie algebra all splitting Cartan algebras are conjugate.
• Over an algebraically closed field, all semisimple Lie algebras are splittable.
• Over a non-algebraically closed field, there exist non-splittable semisimple Lie algebras.[2]
• In a splittable Lie algebra, there may exist Cartan subalgebras that are not splitting.[3]
• Direct sums of splittable Lie algebras and ideals in splittable Lie algebras are splittable.

The split real forms for the complex semisimple Lie algebras are:[6]

• ${\displaystyle A_{n},{\mathfrak {sl}}_{n+1}(\mathbf {C} ):{\mathfrak {sl}}_{n+1}(\mathbf {R} )}$
• ${\displaystyle B_{n},{\mathfrak {so}}_{2n+1}(\mathbf {C} ):{\mathfrak {so}}_{n,n+1}(\mathbf {R} )}$
• ${\displaystyle C_{n},{\mathfrak {sp}}_{n}(\mathbf {C} ):{\mathfrak {sp}}_{n}(\mathbf {R} )}$
• ${\displaystyle D_{n},{\mathfrak {so}}_{2n}(\mathbf {C} ):{\mathfrak {so}}_{n,n}(\mathbf {R} )}$
• Exceptional Lie algebras: ${\displaystyle E_{6},E_{7},E_{8},F_{4},G_{2}}$ have split real forms EI, EV, EVIII, FI, G.

These are the Lie algebras of the split real groups of the complex Lie groups.

Note that for ${\displaystyle {\mathfrak {sl}}}$ and ${\displaystyle {\mathfrak {sp}}}$, the real form is the real points of (the Lie algebra of) the same algebraic group, while for ${\displaystyle {\mathfrak {so}}}$ one must use the split forms (of maximally indefinite index), as the group SO is compact.

• Compact Lie algebra
• Real form
• Split-complex number
• Split orthogonal group