= Lie algebra two-coboundary
{c}
{title2=$B^2(\mathfrak g,k)=\{(x,y)\mapsto t([x,y]):t\in\mathfrak g^*\}$}
For trivial coefficients in a <field> $k$, a Lie algebra two-coboundary is a <bilinear map> $\beta(x,y)=t([x,y])$ for a <linear functional> $t$ in the <dual space> $\mathfrak g^*$. Some conventions put a minus sign in this formula; the resulting <vector subspace> $B^2(\mathfrak g,k)$ is the same. Each such map is a <Lie algebra two-cocycle>: it is an <alternating bilinear form>, and its cyclic expression is $t([[x,y],z]+[[y,z],x]+[[z,x],y])=0$ by the <Jacobi identity>. Replacing a linear section of a <central extension of a Lie algebra> by $s'(x)=s(x)+t(x)z$ changes its <Lie algebra two-cocycle> by $-\beta$. Thus $H^2(\mathfrak g,k)=Z^2(\mathfrak g,k)/B^2(\mathfrak g,k)$ in <Lie algebra cohomology> records exactly the obstruction to a section that is a <Lie algebra homomorphism>.
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