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The adjoint representation is ${\mathrm{ad}}_x(y)=[x,y]$. For a semisimple Lie algebra, its nonzero weights are the roots and its zero-weight space is the Cartan subalgebra.

For a complex semisimple Lie algebra, a Cartan subalgebra is a maximal abelian subalgebra consisting of semisimple elements. It is also called a maximal torus in this setting.

The Lie bracket is the product in a Lie algebra. For matrix Lie algebras it is the commutator $[X,Y]=XY-YX$.

A Lie group is a group that is also a smooth manifold, with smooth multiplication and inversion. Its tangent space at the identity carries a natural Lie algebra structure.

A Lie algebra is nilpotent when its lower central series eventually becomes zero.

Put $\Lambda (t)=I+tX+O(t^2)$ in ${\Lambda }^T\eta \Lambda =\eta$. The coefficient of $t$ is $X^T\eta +\eta X$, so it must vanish. For $X=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ this condition isHence $a=d=0$ and $b=c$. Thus the Lie algebra is one-dimensional with basis

Solved by gpt-5.6-sol high.

The special orthogonal group isIts Lie algebra consists of antisymmetric $5\times 5$ matrices, determined by the entries above the diagonal. Therefore

Solved by gpt-5.6-sol high.

For an ordered basis $(x_i)$ of a Lie algebra $\mathfrak{g}$, the ordered monomials $x_1^{a_1}\cdots x_n^{a_n}$ form a basis of $U(\mathfrak{g})$. In particular, a triangular decomposition $\mathfrak{g}={\mathfrak{n}}^{-}\oplus \mathfrak{t}\oplus {\mathfrak{n}}^{+}$ gives $U(\mathfrak{g})\cong U({\mathfrak{n}}^{-})\otimes U(\mathfrak{t})\otimes U({\mathfrak{n}}^{+})$ as vector spaces.

A principal sl2 subalgebra of a semisimple Lie algebra has semisimple generator $h_{\mathrm{pr}}=2{\rho }^{\vee }$. Restriction to it packages a representation's weights into ordinary ${\mathfrak{sl}}_2$ weight strings.

For a Cartan subalgebra $\mathfrak{t}$, a semisimple Lie algebra decomposes asThe nonzero functionals $\alpha$ are the roots.

A root system is a finite set of nonzero vectors closed under the reflections they define and satisfying the crystallographic integrality condition when it arises from a semisimple Lie algebra.

A finite-dimensional Lie algebra is semisimple when it has no nonzero solvable ideals.

A nonabelian Lie algebra is simple when its only ideals are zero and the entire algebra.