Loop algebra of a simple Lie algebra

Let $\mathfrak{g}$ be a nonzero finite-dimensional complex simple Lie algebra. Here simple means nonabelian with no ideals except $0$ and $\mathfrak{g}$, using the Lie and ideal conventions of Finite semisimple Lie algebras and the symmetric adjoint action. Put $\mathbb{C}[t,t^{-1}]=\{{\sum }_{m\in \mathbb{Z}}{a}_m{t}^m:a_m\in \mathbb{C},\text{ finitely many }{a}_m\mathrm{\ne }0\}$, with multiplication $t^m{t}^n=t^{m+n}$.

The algebraic loop algebra is $L\mathfrak{g}=\mathfrak{g}{\otimes }_{\mathbb{C}}\mathbb{C}[t,t^{-1}]$. Write $x_m=x\otimes t^m$ and define $[x\otimes f,y\otimes q]=[x,y]\otimes fq\mathrm{.}$ This is well-defined on the tensor product because the displayed operation is complex bilinear in each tensor's two entries and respects scalar balancing. On three pure tensors its cyclic Jacobi sum is $([x,[y,z]]+[y,[z,x]]+[z,[x,y]])\otimes fqr=0$. Antisymmetry follows from that in $\mathfrak{g}$ and $fq=qf$. Bilinear and trilinear extension give the identities on all finite sums, including zero. Thus this is a Lie algebra. Only Laurent polynomials occur; no topology or analytic completion is part of the definition.