  This article contains an investigation of the equitable basis for the Lie
algebra sl_2. Denoting this basis by {x,y,z}, we have
  [x,y] = 2x + 2y, [y,z] = 2y + 2z, [z, x] = 2z + 2x.
  One focus of our study is the group of automorphisms G generated by exp(ad
x*), exp(ad y*), exp(ad z*), where {x*,y*,z*} is the basis for sl_2 dual to
{x,y,z} with respect to the trace form (u,v) = tr(uv). We show that G is
isomorphic to the modular group PSL_2(Z). Another focus of our investigation is
the lattice L=Zx+Zy+Zz. We prove that the orbit G(x) equals {u in L |(u,u)=2}.
We determine the precise relationship between (i) the group G, (ii) the group
of automorphisms for sl_2 that preserve L, (iii) the group of automorphisms and
antiautomorphisms for sl_2 that preserve L, and (iv) the group of isometries
for (,) that preserve L. We obtain analogous results for the lattice L*
=Zx*+Zy*+Zz*. Relative to the equitable basis, the matrix of the trace form is
a Cartan matrix of hyperbolic type; consequently,we identify the equitable
basis with the set of simple roots of the corresponding Kac-Moody Lie algebra
g. Then L is the root lattice for g and 1/2L* is the weight lattice, and G(x)
coincides with the set of real roots for g. Using L, L*, and G, we give several
descriptions of the isotropic roots for g and show that each isotropic root has
multiplicity 1. We describe the finite-dimensional sl_2-modules from the point
of view of the equitable basis. In the final section, we establish a connection
between the Weyl group orbit of the fundamental weights of g and Pythagorean
triples.
