Roots of an untwisted affine Lie algebra

For $h\in \mathfrak{h}$, $[h,x_{\alpha }\otimes t^n]=\alpha (h)x_{\alpha }\otimes t^n$, while $d$ acts with eigenvalue $n$ and $c$ acts as zero in the adjoint module. The finite decomposition therefore gives precisely the stated nonzero weights and spaces. Distinct $n$ are distinguished by $d$, and distinct finite weights by $\mathfrak{h}$. The zero-weight space is $\widehat{\mathfrak{h}}$ and is not a root space. Dimensions are one for finite-root modes and $\mathrm{\ell }$ for nonzero Cartan modes.

On the root span define $(\alpha +n\delta ,\beta +m\delta )=(\alpha ,\beta )$ using the positive definite finite-root form. F3's Gram-matrix calculation identifies this with the symmetrized form in F4. Thus $\delta$ is in its radical on the root span, every finite-root mode has positive square, and every simple root, including ${\alpha }_0$, has positive square. The Weyl action preserves this form and fixes $\delta$. Consequently $n\delta$ for $n\mathrm{\ne }0$ cannot be the image of a simple root and is imaginary. This says radical on the root span, not on the full dual Cartan form.

Every root in step 1.1 has simple coordinates of one sign. For $n>0$, $\alpha +n\delta =n{\alpha }_0+(n\theta +\alpha )$ has nonnegative coordinates: F3 gives $\theta +\alpha \in Q_{+}$ and $\theta \in Q_{+}$. The negative-$n$ case follows by negation, and $n=0$ is the finite-root sign rule. Multiples $n\delta$ have the same sign as $n$. In particular, reflecting a positive root not proportional to the reflecting simple root keeps it positive: the reflection changes only that simple coordinate, leaving another positive coordinate unchanged, and its image is a root by F4.

Let $\beta ={\sum }_{i=0}^{\mathrm{\ell }}{b}_i{\alpha }_i$ be a positive root of positive square. The equality $(\beta ,\beta )={\sum }_i{b}_i(\beta ,{\alpha }_i)>0$ yields an $i$ with $(\beta ,{\alpha }_i)>0$. Its positive integral coroot pairing is $(\beta ,{\alpha }_i)\mathrm{/}{d}_i$, with $d_i=({\alpha }_i,{\alpha }_i)\mathrm{/}2>0$. If $\beta$ is proportional to ${\alpha }_i$, F5 gives $\beta ={\alpha }_i$. Otherwise step 2.2 shows that $s_i\beta$ stays positive and has smaller positive integer height. Repeat this descent; height cannot decrease indefinitely, so it reaches a simple root. Reversing the finite reflection word proves $\beta$ real. A negative positive-square root is real as well, since its negative is real and simple roots have real negatives. Thus all $\alpha +n\delta$ are real. Combined with steps 1.1 and 2.1 this proves the complete list and multiplicities. The proof includes rank one and all integer modes; no choice over an infinite index set occurs.