Let
E
E
be a number field and
G
G
a finite group. Let
A
\mathcal {A}
be any
O
E
\mathcal {O}_{E}
-order of full rank in the group algebra
E
[
G
]
E[G]
and
X
X
a (left)
A
\mathcal {A}
-lattice. In a previous article, we gave a necessary and sufficient condition for
X
X
to be free of given rank
d
d
over
A
\mathcal {A}
. In the case that (i) the Wedderburn decomposition
E
[
G
]
≅
⨁
χ
M
χ
E[G] \cong \bigoplus _{\chi } M_{\chi }
is explicitly computable and (ii) each
M
χ
M_{\chi }
is in fact a matrix ring over a field, this led to an algorithm that either gives elements
α
1
,
…
,
α
d
∈
X
\alpha _{1}, \ldots , \alpha _{d} \in X
such that
X
=
A
α
1
⊕
⋯
⊕
A
α
d
X=\mathcal {A}\alpha _{1} \oplus \cdots \oplus \mathcal {A}\alpha _{d}
or determines that no such elements exist. In the present article, we generalise the algorithm by weakening condition (ii) considerably.