An exceptional Lie algebra is a complex simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type. There are exactly five of them: $\mathfrak{g}_{2}$, ${\mathfrak {f}}_{4}$,${\mathfrak {e}}_{6}$, ${\mathfrak {e}}_{7}$, ${\mathfrak {e}}_{8}$; their respective dimensions are 14, 52, 78, 133, 248.
See https://en.wikipedia.org/wiki/Exceptional_Lie_algebra
Usually, given a Lie algebra,
there could be different Lie groups associated to the same given Lie algebras.
It is commonly say the Lie groups of given Lie algebra $\mathfrak{g}_{2}$, ${\mathfrak {f}}_{4}$,${\mathfrak {e}}_{6}$, ${\mathfrak {e}}_{7}$, ${\mathfrak {e}}_{8}$ are $G_2$, $F_4$, $E_6$, $E_7$, $E_8$
However, do we have other and different Lie groups of the same exceptional Lie algebra $\mathfrak{g}_{2}$, ${\mathfrak {f}}_{4}$,${\mathfrak {e}}_{6}$, ${\mathfrak {e}}_{7}$, ${\mathfrak {e}}_{8}$?
For example, the $SO(N)$ and $Spin(N)$ can have the same Lie algebra ${\mathfrak {so}}_{n}$, but different Lie groups because $Spin(N)/(\mathbb{Z}/2)=SO(N)$.