We show that every proper, dense ideal in a
C
∗
C^{*}
-algebra is contained in a prime ideal. It follows that a subset generates a
C
∗
C^{*}
-algebra as a not necessarily closed ideal if and only if it is not contained in any prime ideal.
This allows us to transfer Lie theory results from prime rings to
C
∗
C^{*}
-algebras. For example, if a
C
∗
C^{*}
-algebra
A
A
is generated by its commutator subspace
[
A
,
A
]
[A,A]
as a ring, then
[
[
A
,
A
]
,
[
A
,
A
]
]
=
[
A
,
A
]
[[A,A],[A,A]] = [A,A]
. Further, given Lie ideals
K
K
and
L
L
in
A
A
, then
[
K
,
L
]
[K,L]
generates
A
A
as a not necessarily closed ideal if and only if
[
K
,
K
]
[K,K]
and
[
L
,
L
]
[L,L]
do, and moreover this implies that
[
K
,
L
]
=
[
A
,
A
]
[K,L]=[A,A]
.
We also discover new properties of the subspace generated by square-zero elements and relate it to the commutator subspace of a
C
∗
C^{*}
-algebra.