The Heisenberg subalgebra of an affine Lie algebra

Finite Cartan elements commute. Thus F1 gives the displayed bracket, entirely in $\mathbb{C}c$, proving closure. By F2 each nonzero mode has dimension $\mathrm{\ell }$ and the underlying vectors are precisely the stated Cartan modes. For $m\mathrm{\ne }0$, pairing it with mode $-m$ gives the nondegenerate pairing $mB$ because $B$ is nondegenerate on the finite Cartan.

Let $z=ac+{\sum }_{m\mathrm{\ne }0}{h}_m\otimes t^m$ have finite support and at least one $h_j\mathrm{\ne }0$. Choose $k\in \mathfrak{h}$ with $B(h_j,k)\mathrm{\ne }0$. Its bracket with $k\otimes t^{-j}$ is exactly $jB(h_j,k)c\mathrm{\ne }0$: all other modes have nonopposite degrees and contribute zero. Hence no such $z$ is central. Conversely $ac$ is central by F1. This proves center $\mathbb{C}c$ and nondegeneracy of the alternating form on $\mathcal{H}\mathrm{/}\mathbb{C}c$. If zero modes were included, they would commute with every Cartan mode and enlarge the center by $\mathfrak{h}$. The finite-support zero element and rank-one case require no exception.