  We give a detailed proof of Kolchin's results on differential Galois groups
of strongly normal extensions, in the case where the field of constants is not
necessarily algebraically closed. We closely follow former works due to Pillay
and his co-authors which were written under the assumption that the field of
constant is algebraically closed. In the present setting, which encompasses the
cases of ordered or p-valued differential fields, we find a partial Galois
correspondence and we show one cannot expect more in general. In the class of
ordered differential fields, using elimination of imaginaries in the theory of
closed ordered fields, we establish a relative Galois correspondence for
definable subgroups of the group of differential order automorphisms.
