  The space of tensors of metric curvature type on a Euclidean vector space
carries a two-parameter family of orthogonally invariant commutative
nonassociative multiplications invariant with respect to the symmetric bilinear
form determined by the metric. For a particular choice of parameters these
algebras recover the polarization of the quadratic map on metric curvature
tensors that arises in the work of Hamilton on the Ricci flow. Here these
algebras are studied as interesting examples of metrized commutative algebras
and in low dimensions they are described concretely in terms of nonstandard
commutative multiplications on self-adjoint endomorphisms.
  The algebra of curvature tensors on a three-dimensional Euclidean vector
space is shown isomorphic to an orthogonally invariant deformation of the
standard Jordan product on three by three symmetric matrices. This algebra is
characterized up to isomorphism in terms of purely algebraic properties of its
idempotents and the spectra of their multiplication operators.
  On a vector space of dimension at least four, the subspace of Weyl
(Ricci-flat) curvature tensors is a subalgebra for which the multiplication
endomorphisms are trace-free and the Killing type trace-form is a multiple of
the nondegenerate invariant metric. This subalgebra is simple when the
Euclidean vector space has dimension greater than four. In the presence of a
compatible complex structure, the analogous result is obtained for the
subalgebra of K\"ahler Weyl curvature tensors. It is shown that the
anti-self-dual Weyl tensors on a four-dimensional vector space form a simple
five-dimensional ideal isometrically isomorphic to the trace-free part of the
Jordan product on trace-free $3 \times 3$ symmetric matrices.
