  The purpose of this paper is to synthesize the approaches taken by
Chatterjee-Meckes and Reinert-R\"ollin in adapting Stein's method of
exchangeable pairs for multivariate normal approximation. The more general
linear regression condition of Reinert-R\"ollin allows for wider applicability
of the method, while the method of bounding the solution of the Stein equation
due to Chatterjee-Meckes allows for improved convergence rates. Two abstract
normal approximation theorems are proved, one for use when the underlying
symmetries of the random variables are discrete, and one for use in contexts in
which continuous symmetry groups are present. The application to runs on the
line from Reinert-R\"ollin is reworked to demonstrate the improvement in
convergence rates, and a new application to joint value distributions of
eigenfunctions of the Laplace-Beltrami operator on a compact Riemannian
manifold is presented.
