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A Lie algebra is abelian when every Lie bracket vanishes.

The commutator of two elements of an associative algebra is $[A,B]=AB-BA$. Matrix Lie algebras use the commutator as their Lie bracket.

A Lie algebra over a field is a vector space $\mathfrak{g}$ with a bilinear operation $[x,y]$, called the Lie bracket, that is alternating and satisfies the Jacobi identity.

The Nijenhuis tensor of an almost complex manifold isIt measures the failure of the $(0,1)$ tangent distribution to be closed under the Lie bracket.

For the trace trilinear form of a Lie algebra representation, the trace of a commutator vanishes:Taking $X=T_b$, $Y=T_a$, $Z=T_c$, $W=T_d$ and expanding each Lie bracket givesSince $f_a{}^{bc}$ is antisymmetric in $b,c$,Raising indices with the inverse Killing form gives $f_a{}^{bc}{f}_{bc}{}^e={\delta }_a{}^e$ in the stated normalization. Using $H(T_e,T_d)=-\mu {\delta }_{ed}$ proves

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