Twisted loop algebra from a diagram automorphism

Let $\sigma$ be an automorphism of the finite simple $\mathfrak{g}$ induced by a Dynkin-diagram permutation of its fixed simple generators, of finite order $r\ge 1$. Fix a primitive $r$th root of unity $\zeta \in \mathbb{C}$. Write ${\mathfrak{g}}_j=\{x:\sigma (x)={\zeta }^{j}x\}$, with indices modulo $r$. The polynomial $X^r-1$ has distinct roots, so its annihilation of $\sigma$ gives $\mathfrak{g}={⨁}_{j<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>r}{\mathfrak{g}}_j$.

In the loop realization of Degree derivation and full untwisted affine algebra, define $\tau (x\otimes t^m)={\zeta }^{-m}\sigma (x)\otimes t^m,\tau (c)=c,\tau (d)=d\mathrm{.}$ The twisted loop algebra is the fixed subalgebra $L(\mathfrak{g},\sigma )={⨁}_{m\in \mathbb{Z}}{\mathfrak{g}}_{m<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>r}\otimes t^m\mathrm{.}$ Its central/full extensions here mean the fixed subalgebras $L(\mathfrak{g},\sigma )\oplus \mathbb{C}c$ and $L(\mathfrak{g},\sigma )\oplus \mathbb{C}c\oplus \mathbb{C}d$ in that same normalization.

For well-definedness, $\sigma$ preserves the normalized Killing form: conjugation intertwines the finite adjoint operators, preserving their product traces, and hence also the fixed scalar normalization. Thus the loop part of each bracket is preserved. A nonzero central coefficient has $m+n=0$, so its scalar factor under $\tau$ is ${\zeta }^{-m-n}=1$; the central term is preserved as well. The degree action is preserved since $\tau$ does not change $m$. Therefore $\tau$ is a Lie automorphism, with inverse obtained from ${\sigma }^{-1}$ and ${\zeta }^m$. If two elements are fixed, so is their bracket. Finally the degree-$m$ fixed condition is exactly $\sigma (x)={\zeta }^{m}x$, proving the displayed description. Brackets satisfy $[{\mathfrak{g}}_j,{\mathfrak{g}}_k]\subseteq {\mathfrak{g}}_{j+k}$ directly by applying $\sigma$.

For $r=1$ this is the untwisted construction. All mode sums are finite; some eigenspaces may be zero. This defines the fixed-loop objects, without classifying twisted affine diagrams or changing the central/degree scaling convention.