  In this paper, we firstly establish Composition-Diamond lemma for
$\Omega$-algebras. We give a Gr\"{o}bner-Shirshov basis of the free $L$-algebra
as a quotient algebra of a free $\Omega$-algebra, and then the normal form of
the free $L$-algebra is obtained. We secondly establish Composition-Diamond
lemma for $L$-algebras. As applications, we give Gr\"{o}bner-Shirshov bases of
the free dialgebra and the free product of two $L$-algebras, and then we show
four embedding theorems of $L$-algebras: 1) Every countably generated
$L$-algebra can be embedded into a two-generated $L$-algebra. 2) Every
$L$-algebra can be embedded into a simple $L$-algebra. 3) Every countably
generated $L$-algebra over a countable field can be embedded into a simple
two-generated $L$-algebra. 4) Three arbitrary $L$-algebras $A$, $B$, $C$ over a
field $k$ can be embedded into a simple $L$-algebra generated by $B$ and $C$ if
$|k|\leq \dim(B*C)$ and $|A|\leq|B*C|$, where $B*C$ is the free product of $B$
and $C$.
