Let
E
E
be a central extension of the form
0
→
V
→
G
→
W
→
0
0 \to V \to G \to W \to 0
where
V
V
and
W
W
are elementary abelian
2
2
-groups. Associated to
E
E
there is a quadratic map
Q
:
W
→
V
Q: W \to V
, given by the
2
2
-power map, which uniquely determines the extension. This quadratic map also determines the extension class
q
q
of the extension in
H
2
(
W
,
V
)
H^2(W,V)
and an ideal
I
(
q
)
I(q)
in
H
2
(
G
,
Z
/
2
)
H^2(G, \mathbb {Z} /2)
which is generated by the components of
q
q
. We say that
E
E
is Bockstein closed if
I
(
q
)
I(q)
is an ideal closed under the Bockstein operator. We find a direct condition on the quadratic map
Q
Q
that characterizes when the extension is Bockstein closed. Using this characterization, we show for example that quadratic maps induced from the fundamental quadratic map
Q
g
l
n
:
g
l
n
(
F
2
)
→
g
l
n
(
F
2
)
Q_{\mathfrak {gl}_n}: \mathfrak {gl}_n (\mathbb {F}_2)\to \mathfrak {gl}_n (\mathbb {F}_2)
given by
Q
(
A
)
=
A
+
A
2
Q(\mathbb {A})= \mathbb {A} +\mathbb {A} ^2
yield Bockstein closed extensions. On the other hand, it is well known that an extension is Bockstein closed if and only if it lifts to an extension
0
→
M
→
G
~
→
W
→
0
0 \to M \to \widetilde {G} \to W \to 0
for some
Z
/
4
[
W
]
\mathbb {Z} /4[W]
-lattice
M
M
. In this situation, one may write
β
(
q
)
=
L
q
\beta (q)=Lq
for a “binding matrix”
L
L
with entries in
H
1
(
W
,
Z
/
2
)
H^1(W, \mathbb {Z}/2)
. We find a direct way to calculate the module structure of
M
M
in terms of
L
L
. Using this, we study extensions where the lattice
M
M
is diagonalizable/triangulable and find interesting equivalent conditions to these properties.