Given a root system
R
R
and the corresponding finite reflection group
W
W
let
Hom
(
W
,
Z
^
2
)
\operatorname {Hom}(W,\,\widehat {\mathbb Z}_2)
be the group of homomorphisms from
W
W
into
Z
^
2
\widehat {\mathbb Z}_2
, where
Z
^
2
=
{
1
,
−
1
}
\widehat {\mathbb Z}_2=\{1,-1\}
with multiplication. We propose a procedure of constructing subroot systems of
R
R
by using homomorphisms
η
∈
Hom
(
W
,
Z
^
2
)
\eta \in \operatorname {Hom}(W,\,\widehat {\mathbb Z}_2)
. This construction is next used for establishing a relation between concepts of total positivity and
η
\eta
-total positivity.