A Lie algebra $\mathfrak{g}$ has a central extension $\mathfrak{𝔤}_{\mu}$ with central charge $\mu$. Is there a family of Lie algebras $\mathfrak{g}_{\alpha\mu}$, for which $\mathfrak{g}_{\alpha\mu} \cong \mathfrak{g}_{\alpha} \oplus \langle \mu\rangle$ for $\alpha \neq 0$ (where $\langle\mu\rangle$ is the one-dimensional Lie algebra generated by $\mu$) such that $\mathfrak{g}_{\alpha\mu} \to \mathfrak{g}_{μ}$ and $\mathfrak{g}_{\alpha} \to \mathfrak{g}$ as $\alpha \to 0$?

**Example**

$\mathfrak{g}: (X, Y, Z : [Y,Z] = 0, [Z,X] = Z, [X,Y] = Y)$

$\mathfrak{g}_{\mu} : (X, Y, Z, \mu : [Y,Z] = \mu, [Z,X] = Z, [X,Y] = Y, [\mu,X]=[\mu,Y]=[\mu,Z] = 0)$

$\mathfrak{g}_{\alpha\mu} : (X, Y, Z, \mu : [Y,Z] = \mu + \alpha X, [Z,X] = Z, [X,Y] = Y, [\mu,X]=[\mu,Y]=[\mu,Z] = 0)$

$\mathfrak{g}_{\alpha} : (W, Y, Z: [Y,Z] = \alpha W, [Z,W] = Z, [W,Y] = Y)$

with the decomposition $\mathfrak{g}_{\alpha\mu} \cong \mathfrak{g}_{\alpha}\oplus \langle\mu\rangle$ for $\alpha \neq 0$ given by $W = X + \mu/\alpha$.