Let $L$ be a Lie algebra over $\mathbb{Z}$ constructed from a root lattice $R$. It is well-known that if $R=A_1$, then $L \cong \mathfrak{sl}_2$ and this is widely used example in many books on Lie algebras. However, I have never seen a strict proof of this fact. How can we actually show the isomorphism?
2026-03-26 20:40:03.1774557603
$\mathfrak{sl}_2$ has the root lattice of type $A_1$.
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