In a ring
R
with involution whose symmetric elements
S
are central, the skew-symmetric elements
K
form a Lie algebra over the commutative ring
S
. The classification of such rings which are
2
-torsion free is equivalent to the classification of Lie algebras
K
over
S
equipped with a bilinear form
f
that is symmetric, invariant and satisfies
[
[
x
,
y
]
,
z
]
=
f
(
y
,
z
)
x
−
f
(
z
,
x
)
y
. If
S
is a field of char
≠
2
,
f
≠
0
and
dim
K
>
1
then
K
is a semisimple Lie algebra if and only if
f
is nondegenerate. Moreover, the derived algebra
K
′
is either the pure quaternions over
S
or a direct sum of mutually orthogonal abelian Lie ideals of
dim
≤
2
.