Consider the probability distribution $\mathcal{N}_n$ on $\mathbb{R}^n$ whose density is $$(2\pi)^{-n/2}e^{-\frac{1}{2}||\vec{x}||^2} = \prod_{i=1}^n \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x_i^2}$$
This multivariate normal distribution has symmetry group $O(n)$. But the family of these distributions has two additional symmetry properties:
(a) If $X$ has distribution $\mathcal{N}_n$, and $\pi$ is an orthogonal projection onto an $k$ dimensional subspace, then $\pi \circ X$ has distribution $\mathcal{N}_k$.
(b) If $X$ and $Y$ are independent random variables with distribution $\mathcal{N}_k$ and $\mathcal{N}_j$, then $X \oplus Y$ (the random variable on $\mathbb{R}^{k+j}$) has probability distribution $\mathcal{N}_{k+j}$
My question: is $(\mathcal{N}_n : n \in \mathbb{N})$ the only family of probability distributions (up to rescaling) which has properties (a) and (b)?