For an $NIG(\alpha, \beta, \mu, \delta)$ distributed random variable $X$, it is established (see e.g. here)) that $X$ can be constructed as a normal mean-variance mixture of a Gaussian random variable $U\sim N(0,1)$ subordinated by an Inverse Gaussian distributed random variable $Z\sim IG(\delta,\gamma)$ with a scaled and shifted version of $Z$, \begin{equation} X = \mu + \beta Z + \sqrt{Z} U, \end{equation} where $\alpha = \sqrt{\beta^2 + \gamma^2}$, the IG density $f_Z$ of $Z$ is given by \begin{eqnarray} f_Z(z) &=& \frac{\delta}{\sqrt{2\pi z^3}} e^{\delta\gamma} \exp\left[-\frac{1}{2} (\delta^2/z + \gamma^2 z) \right] \\ &=& \frac{\delta}{\sqrt{2\pi z^3}} \exp\left[-\frac{1}{2z} (\gamma z -\delta)^2 \right] \; \end{eqnarray} and we have $\sqrt{Z} U \sim N(0,Z)$ due to the properties of the normal distribution.
The density $f_X$ of the $NIG(\alpha, \beta, \mu, \delta)$ distribution is then given by \begin{equation} f_X(x) = \frac{\alpha \delta}{\pi}\exp\left[ \delta \sqrt{\alpha^2-\beta^2}-\beta\mu \right] \, \frac{K_1(\alpha \sqrt{\delta^2 + (x-\mu)^2} )}{\sqrt{\delta^2 + (x-\mu)^2}}, \end{equation} where $K_1$ is the modified Bessel function of the third kind.
I found a claim that $X$ can also be constructed as $X_1$ by using $Z_1$ distributed $IG(1,b)$ with \begin{equation} b = \delta \sqrt{\alpha^2-\beta^2} \, \end{equation} setting \begin{equation} X_1 = \mu + \beta \delta^2 Z_1 + \delta \sqrt{Z_1} U \; ; \end{equation} then $X_1$ would also have a $NIG(\alpha, \beta, \mu, \delta)$ distribution.
I do not see how this claim can be proven (e.g. from the above). Is it due to a different parametrization? I tried to compute the Laplace exponent $L_{X_1}(u)= \log \mathbb{E}\left[e^{u{X_1}}\right]$ of ${X_1}$ and compare it with $L_X$ of $X$; the latter has been shown to be \begin{equation} L_X(u) = \log \mathbb{E}\left[e^{uX}\right] = \delta \left( \sqrt{\alpha^2-\beta^2} - \sqrt{\alpha^2 - (\beta+u)^2} \right) + \mu u \; ; \end{equation} then I compute $L_{X_1}$, e.g. via splitting $X_1$ into $A = \mu +\beta \delta^2 Z_1$ and $B = \delta \sqrt{Z_1} U$ and using \begin{equation} \mathbb{E}\left[e^{u{X_1}}\right] = \mathbb{E}\left[e^{uA} e^{uB} \right] = \mathbb{E}\left[e^{uA}\right] e^{L_{Z_1}(\phi(u;0,\delta^2))} \;, \end{equation} where $L_{Z_1}(u) = L_{IG}(u;1,b)$ with $L_{IG}(u;a,c)$ the Laplace exponent of an $IG(a,c)$-distributed random variable, \begin{equation} L_{IG}(u;a,c) = a ( c - \sqrt{c^2-2u}) \;, \end{equation} and $\phi(u;m,s^2)$ being the moment generating function of the normal distribution with mean $m$ and variance $s^2$, \begin{equation} \phi(u;m,s^2) = m u + \frac{1}{2} s^2 u^2 \; ; \end{equation} however I get conflicting results, containing $1/\delta$ in the front of $L_{ X_1}$ for example.
What is missing or wrong? I would be grateful for any help. Thanks in advance!