  For any $n$-dimensional 3-Lie algebra $A$ over a field of characteristic zero
with an involutive derivation $D$, we investigate the structure of the 3-Lie
algebra $B_1=A\ltimes_{ad^*} A^* $ associated with the coadjoint representation
$(A^*, ad^*)$. We then discuss the structure of the dual 3-Lie algebra $B_2$ of
the local cocycle 3-Lie bialgebra $(A\ltimes_{ad^*} A^*, \Delta)$. By means of
the involutive derivation $D$, we construct the $4n$-dimensional Manin triple
$(B_1\oplus B_2,$ $ [ \cdot, \cdot, \cdot]_1,$ $ [ \cdot, \cdot, \cdot]_2,$ $
B_1, B_2)$ of 3-Lie algebras, and provide concrete multiplication in a special
basis $\Pi_1\cup\Pi_2$. We also construct a sixteen dimensional Manin triple
$(B, [ \cdot, \cdot, \cdot])$ with $\dim B^1=12$ using an involutive derivation
on a four dimensional 3-Lie algebra $A$ with $\dim A^1=2$.
