The connected covers of the classifying space
B
O
BO
induce a decreasing filtration
{
B
n
}
\{ {B_n}\}
of
H
∗
(
B
O
;
Z
/
2
)
{H_{\ast }}(BO;\,Z/2)
by sub-Hopf algebras over the Steenrod algebra
A
A
. We describe a multiplicative grading on
H
∗
(
B
O
;
Z
/
2
)
{H_{\ast }}(BO;\,Z/2)
inducing a direct sum splitting of
B
n
{B_n}
over
A
n
{A_n}
, where
{
A
n
}
\{ {A_n}\}
is the usual (increasing) filtration of
A
A
. The pieces in the splittings are finite, and the grading extends that of
H
∗
Ω
2
S
3
{H_{\ast }}{\Omega ^2}{S^3}
which splits it into Brown-Gitler modules. We also apply the grading to the Thomifications
{
M
n
}
\{ {M_n}\}
of
{
B
n
}
\{ {B_n}\}
, where it induces splittings of the corresponding cobordism modules over the entire Steenrod algebra. These generalize algebraically the previously known topological splittings of the connective cobordism spectra
M
O
MO
,
M
S
O
MSO
and
M
S
p
i
n
M\,Spin
.