  Constructive algorithms, requiring no more than $2\times 2$ matrix
manipulations, are provided for finding the entries of the positive definite
factor in the polar decomposition of matrices in sixteen groups preserving a
bilinear form in dimension four, including the Lorentz and symplectic groups.
This is used to find quaternionic representations for these groups analogous to
that for the special orthogonal group.This is achieved by first characterizing
positive definite matrices in these groups. For the groups whose signature
matrix is a symmetric matrix in the quaternion tensor basis for 4x4 matrices, a
completion procedure based on this observation leads to said computation of the
polar decomposition, while for the Lorentz group this is achieved by passage to
its double cover. Amongst byproducts we mention an elementary and constructive
proof showing that positive definite matrices in such groups belong to the
connected component of the identity.
