Editing Semisimple Lie algebra (section)

== Examples ==
As noted in [[#Structure]], semisimple [[Lie algebra]]s over <math>\mathbb{C}</math> (or more generally an algebraically closed field of characteristic zero) are classified by the root system associated to their Cartan subalgebras, and the root systems, in turn, are classified by their Dynkin diagrams.
Examples of semisimple Lie algebras, the [[classical Lie algebra]]s, with notation coming from their [[Dynkin diagram]]s, are:
* <math>A_n:</math> <math>\mathfrak {sl}_{n+1}</math>, the [[special linear Lie algebra]].
* <math>B_n:</math> <math>\mathfrak{so}_{2n+1}</math>, the odd-dimensional [[special orthogonal Lie algebra]].
* <math>C_n:</math> <math>\mathfrak {sp}_{2n}</math>, the [[symplectic Lie algebra]].
* <math>D_n:</math> <math>\mathfrak{so}_{2n}</math>, the even-dimensional [[special orthogonal Lie algebra]] (<math>n>1</math>).
The restriction <math>n>1</math> in the <math>D_n</math> family is needed because <math>\mathfrak{so}_{2}</math> is one-dimensional and commutative and therefore not semisimple.

These Lie algebras are numbered so that ''n'' is the [[rank (Lie algebra)|rank]]. Almost all of these semisimple Lie algebras are actually simple and the members of these families are almost all distinct, except for some collisions in small rank. For example <math>\mathfrak{so}_{4} \cong \mathfrak{so}_{3} \oplus \mathfrak{so}_{3} </math> and <math>\mathfrak{sp}_{2} \cong \mathfrak{so}_{5}</math>. These four families, together with five exceptions ([[E6 (mathematics)|E<sub>6</sub>]], [[E7 (mathematics)|E<sub>7</sub>]], [[E8 (mathematics)|E<sub>8</sub>]], [[F4 (mathematics)|F<sub>4</sub>]], and [[G2 (mathematics)|G<sub>2</sub>]]), are in fact the ''only'' simple Lie algebras over the complex numbers.