Let
be a neutral Tannakian category over a field of characteristic zero with unit object
, and equipped
with a filtration
similar to the weight filtration on mixed motives. Let
be an
object of
,
and
the Lie algebra of the kernel of the natural surjection from the fundamental group of
to the fundamental
group of
.
A result of Deligne gives a characterization of
in terms of the
extensions
: it states
that
is the smallest
subobject of
such that the sum of the aforementioned extensions, considered as extensions of
by
, is the pushforward
of an extension of
by
.
We study each of the above-mentioned extensions individually in relation to
.
Among other things, we obtain a refinement of Deligne’s result,
where we give a sufficient condition for when an individual extension
is the pushforward
of an extension of
by
. In
the second half of the paper, we give an application to mixed motives whose
unipotent radical of the motivic Galois group is as large as possible (i.e., with
).
Using Grothendieck’s formalism of
extensions panachées we prove a classification
result for such motives. Specializing to the category of mixed Tate motives
we obtain a classification result for 3-dimensional mixed Tate motives over
with
three weights and large unipotent radicals.