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Over a field $k$, the trace-zero $2\times 2$ matrices form the special linear Lie algebra ${\mathfrak{sl}}_2(k)$. The matrix trace of a commutator is zero, so this is a Lie subalgebra of the general linear Lie algebra. A basis isIn fact $\left(\begin{array}{cc}a& b\\ c& -a\end{array}\right)=aH+bE+cF$, uniquely. The individual products areSubtracting in the stated order givesFor characteristic different from two, this is the standard sl2 triple; see Etingof, Section 11.4. The products remain valid in characteristic two, but there $H=I$ is central and $2E=2F=0$, so the familiar characteristic zero structure changes.

The following table gives all nine basis brackets. Each entry is $[\text{row},\text{column}]$; the diagonal vanishes by alternation, and exchanging row and column reverses the sign.

Bilinearity now computes any commutator, for exampleThe three structure constants of a Lie algebra relations encode all of this multiplication. They also show why the sl2 triple is useful in Lie algebra representations: the actions of $E$ and $F$ move between eigenvalues of the action of $H$ in steps of two.

The special linear group acts on determinant-one real symmetric matrices by matrix congruence. Symmetry and determinant are preserved, and signatures are preserved by Sylvester's law of inertia. The determinant-one set is a regular level set of dimension $n(n+1)/2-1$, with tangent directions $C=C^T$ satisfying $\mathrm{tr}(B^{-1}C)=0$. Its infinitesimal left-action fields are $XB+B{X}^T$; converting them to a Lie algebra representation requires the infinitesimal left-action sign convention.

A subrepresentation is an invariant subspace of a group representation or Lie algebra representation, with the restricted action.

Over every field, including characteristic two, there is a natural isomorphismThe middle summand maps $v\otimes w$ to the mixed product $vw$. Monomial bases prove bijectivity, and the Leibniz rule makes the map equivariant for Lie algebra representations.

The tensor product of Lie algebra representations has action $x(v\otimes w)=xv\otimes w+v\otimes xw$. The Lie algebra representation identity follows because operators on the two different tensor factors commute. This action descends to exterior powers and symmetric powers.

For $n\ge 2$, the defining Lie algebra representation $V$ of ${\mathfrak{sp}}_{2n}$ has the displayed tensor product decomposition. The crystal of the defining symplectic representation and the crystal tensor-product rule give exactly these three highest vertices, each occurring once. The symmetric square is $L(2{\omega }_1)$ of dimension $n(2n+1)$. The exterior square splits into its invariant inverse-form line and the primitive exterior square $L({\omega }_2)$ of dimension $n(2n-1)-1$. Their dimensions sum to $4n^2$. At $n=1$, the exterior square is only the invariant line, giving $L(2{\omega }_1)\oplus L(0)$.

For a finite-dimensional Lie algebra representation $\rho :\mathfrak{g}\to \mathfrak{gl}(V)$, its trace form is the symmetric bilinear formIt is invariant: $([x,y],z{)}_V=(x,[y,z]{)}_V$.

A weight basis of a Lie algebra representation is a basis consisting of weight vectors for a chosen Cartan subalgebra. When the Cartan action is diagonalizable, choose a basis separately in every weight space and combine them. This makes weight multiplicities directly visible and gives diagonal matrices for Cartan elements.

A weight-space decomposition expresses a Lie algebra representation as a direct sum of simultaneous eigenspaces for a Cartan subalgebra. Each summand is the weight space for its weight. In complex finite-dimensional representations of a semisimple Lie algebra, the simple coroots act as commuting diagonalizable operators, giving this decomposition.