There are two well-known ways of describing elements of the rotation group SO$\left(m\right)$. First, according to the Cartan–Dieudonné theorem, every rotation matrix can be written as an even number of reflections. And second, they can also be expressed as the exponential of some anti-symmetric matrix.

In this paper, we study similar descriptions of a group of rotations SO${}_0$ in the superspace setting. This group can be seen as the action of the functor of points of the orthosymplectic supergroup $\text{OSp}\left(m\right|2n)$ on a Grassmann algebra. While still being connected, the group SO${}_0$ is thus no longer compact. As a consequence, it cannot be fully described by just one action of the exponential map on its Lie algebra. Instead, we obtain an Iwasawa-type decomposition for this group in terms of three exponentials acting on three direct summands of the corresponding Lie algebra of supermatrices.

At the same time, SO${}_0$ strictly contains the group generated by super-vector reflections. Therefore, its Lie algebra is isomorphic to a certain extension of the algebra of superbivectors. This means that the Spin group in this setting has to be seen as the group generated by the exponentials of the so-called extended superbivectors in order to cover SO${}_0$. We also study the actions of this Spin group on supervectors and provide a proper subset of it that is a double cover of SO${}_0$. Finally, we show that every fractional Fourier transform in n bosonic dimensions can be seen as an element of this spin group.

描述旋转组SO的元素有两种众所周知的方法$（米）$。首先，根据Cartan–Dieudonné定理，每个旋转矩阵都可以写成偶数个反射。其次，它们也可以表示为一些反对称矩阵的指数。

在本文中，我们研究了一组旋转SO的类似描述${}_0$在超空间设置中。该组可以看作是正统超群点的函子的作用$\text{操作系统}（米|2ñ）$在Grassmann代数上。在仍处于连接状态时，SO组${}_0$因此不再紧凑。结果，仅靠其李代数上的指数映射的一个动作就不能完全描述它。取而代之的是，我们根据作用在超矩阵李代数的三个直接和上的三个指数，获得了该组的Iwasawa型分解。

同时，所以${}_0$严格包含由超向量反射生成的组。因此，它的李代数与超双向量的代数的一定扩展是同构的。这意味着在这种情况下，自旋组必须视为由所谓的扩展超双矢量的指数生成的组才能覆盖SO。${}_0$。我们还研究了此Spin组对超向量的作用，并提供了它的适当子集，该子集是SO的双重覆盖${}_0$。最后，我们表明，n个玻色子维中的每个分数阶傅里叶变换都可以看作是该自旋群的一个元素。