  Let G be a complex semisimple linear algebraic group and let Pet be the
associated Peterson variety in the flag variety G/B. The main theorem of this
note gives an efficient presentation of the equivariant cohomology ring
H^*_S(Pet) of the Peterson variety as a quotient of a polynomial ring by an
ideal J generated by quadratic polynomials, in the spirit of the Borel
presentation of the cohomology of the flag variety. Here the group S \cong
\mathbb{C}^* is a certain subgroup of a maximal torus T of G. Our description
of the ideal J uses the Cartan matrix and is uniform across Lie types. In our
arguments we use the Monk formula and Giambelli formula for the equivariant
cohomology rings of Peterson varieties for all Lie types, as obtained in the
work of Drellich. Our result generalizes a previous theorem of
Fukukawa-Harada-Masuda, which was only for Lie type A.
