= Lie-Poisson bracket
{c}
{title2=$\{F,G\}(\ell)=\ell([dF_\ell,dG_\ell])$}
= Linear Poisson bracket
{synonym}
The <dual space> $\mathfrak g^*$ of a finite-dimensional <Lie algebra> has this canonical <Poisson bracket>. For a basis $v^i$, linear coordinates $x^i(\ell)=\ell(v^i)$ give $\{x^i,x^j\}=c^{ij}_kx^k$. The Lie algebra's <Jacobi identity> is precisely the <coordinate Jacobi condition for a Poisson bivector>. This construction is intrinsic on the dual space; writing the same formula in an arbitrary manifold chart does not by itself provide a compatible global atlas.
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