Let
H
H
be a pointed Hopf algebra with abelian coradical. Let
A
⊇
B
A\supseteq B
be left (or right) coideal subalgebras of
H
H
that contain the coradical of
H
H
. We show that
A
A
has a PBW basis over
B
B
, provided that
H
H
satisfies certain mild conditions. In the case that
H
H
is a connected graded Hopf algebra of characteristic zero and
A
A
and
B
B
are both homogeneous of finite Gelfand-Kirillov dimension, we show that
A
A
is a graded iterated Ore extension of
B
B
. These results turn out to be conceptual consequences of a structure theorem for each pair
S
⊇
T
S\supseteq T
of homogeneous coideal subalgebras of a connected graded braided bialgebra
R
R
with braiding satisfying certain mild conditions. The structure theorem claims the existence of a well-behaved PBW basis of
S
S
over
T
T
. The approach to the structure theorem is constructive by means of a combinatorial method based on Lyndon words and braided commutators, which is originally developed by V. K. Kharchenko [Algebra Log. 38 (1999), pp. 476–507, 509] for primitively generated braided Hopf algebras of diagonal type. Since in our context we don’t priorilly assume
R
R
to be primitively generated, new methods and ideas are introduced to handle the corresponding difficulties, among others.