Let $\nu$ be a symmetric $\alpha$-stable Levy measure on $\mathbb{R}^d$, $\alpha \in (0,2]$ (if you like, let $d=2$). Since the density of $\nu$ is radial, the image measure of $\nu$ under $x \mapsto \langle x, v\rangle_d$, where $\langle \cdot, \cdot \rangle_d$ denotes the usual inner product on $\mathbb{R}^d$, is the same for any $v, w \in \mathbb{R}^d$, if $|v| = |w|$. So, without loss, consider $v = e_1$, i.e. the associated map is the projection on the first coordinate. By slight abuse of notation, denote this map by $e_1$.
My question is: What can we say about $\nu \circ e_1^{-1}$? Is it a one-dim. $\alpha$-stable Levy measure, possible with a density with an additional dimension-dependent constant and a different $\alpha$?