In geometric algebra, the rotor group is the Lie group obtained by restricting ${\textrm{Spin}\left(r,s\right)}$ to elements whose reverse is their inverse, i.e. elements which satisfy ${U\widetilde{U}=1}$. For ${n>2}$ this restriction results in the identity component, i.e the rotor group is just ${\textrm{Spin}^{e}}$. Thus in this context for ${n>2}$ we can write ${R_{U}^{e}(v)=Uv\widetilde{U}}$. This rotation operator can also be applied to any multivector ${A=\Sigma\left\langle A\right\rangle _{k}}$ to yield the “rotated” multivector ${UA\widetilde{U}}$. Under this operation, each ${k}$-blade, consisting of the exterior product of ${k}$ vectors, is replaced with the exterior product of ${k}$ rotated vectors.

The vectors in any space-like plane in a vector space can be identified with the complex numbers via the representation of the isomorphism ${C_{0}\left(2,0\right)\cong C\left(0,1\right)\cong\mathbb{C}}$ effected by ${\hat{e}_{1}\hat{e}_{2}\equiv\Omega\rightarrow i}$, where ${i}$ is the unit vector in ${C\left(0,1\right)}$ identified with the imaginary unit in ${\mathbb{C}}$. A vector ${v\equiv v^{1}\hat{e}_{1}+v^{2}\hat{e}_{2}\in C\left(2,0\right)}$ is represented by ${v_{\mathrm{E}}\equiv\hat{e}_{1}v=v^{1}+v^{2}\hat{e}_{1}\hat{e}_{2}}$ in the even subalgebra ${C_{0}\left(2,0\right)}$, and therefore by ${v_{\mathbb{C}}=v^{1}+iv^{2}}$ in ${\mathbb{C}}$, where the choice of ${\hat{e}_{1}}$ thus defines the real axis. Complex conjugation is then the reflection across the imaginary axis ${\hat{e}_{1}v_{\mathrm{E}}\hat{e}_{1}=v\hat{e}_{1}=\tilde{v}_{\mathrm{E}}=v^{1}+v^{2}\hat{e}_{2}\hat{e}_{1}=v^{1}-iv^{2}=v_{\mathbb{C}}^{*}}$, which is also reversion in ${C_{0}\left(2,0\right)}$. The complex inner product is ${\left\langle v_{\mathbb{C}},w_{\mathbb{C}}\right\rangle _{\mathbb{C}}=v_{\mathbb{C}}^{*}w_{\mathbb{C}}=\tilde{v}_{\mathrm{E}}w_{\mathrm{E}}=\left\langle v,w\right\rangle _{\mathbb{R}}}$. Note that multiplication by the imaginary unit in ${\mathbb{C}}$ is represented by right Clifford multiplication by ${\Omega}$ in ${C_{0}\left(2,0\right)}$: ${v_{\mathrm{E}}\Omega=\hat{e}_{1}v\Omega=v^{1}\Omega-v^{2}=iv^{1}-v^{2}=iv_{\mathbb{C}}}$. This means that exponential rotations must also act from the right in ${C_{0}\left(2,0\right)}$, and since ${\Omega}$ anti-commutes with vectors, both operations from the left reverse sign: ${e^{\Omega\theta}v_{\mathrm{E}}=e^{\Omega\theta}\left(\hat{e}_{1}v\right)=\hat{e}_{1}\left(e^{-\Omega\theta}v\right)=\left(e^{-\Omega\theta}v\right)_{\mathrm{E}}=e^{-i\theta}v_{\mathbb{C}}}$.

The representation of the isomorphism ${C_{0}\left(1,3\right)\cong C\left(3,0\right)}$ effected by ${\hat{e}_{i}\hat{e}_{0}\rightarrow\sigma_{i}}$ is sometimes called a space-time split in geometric algebra, since the resulting basis of ${C(3,0)}$ reflects (and depends upon) the particular chosen orthonormal basis ${\hat{e}_{i}}$ of ${C_{0}(1,3)}$. An event ${x\in C(1,3)}$ with spacetime coordinates ${x^{\mu}\hat{e}_{\mu}}$ is represented by ${x\hat{e}_{0}=x^{0}+x^{i}\sigma_{i}}$ in ${C(3,0)}$; such a linear combination of scalar and vector in ${C(3,0)}$ is then called a paravector (although this term is sometimes used differently). This scheme can be used to treat relativistic physics in a condensed manner. Note that a space-time split “preserves” the scalar and pseudo-scalar basis: ${I\rightarrow I}$ and ${\Omega\rightarrow\Omega}$. If spacetime is instead represented by the “mostly pluses” signature algebra ${C(3,1)}$, ${-x\hat{e}_{0}}$ can be used as the space-time split with ${\hat{e}_{0}\hat{e}_{i}\rightarrow\sigma_{i}}$ in order to make the signs come out right.

There is also an interesting alternative to the standard definition of Dirac and Weyl spacetime spinors (as vectors acted on by a faithful complex representation of ${\textrm{Spin}(3,1)^{e}}$), which instead considers these spinors as elements of the Clifford algebra associated with space. The Dirac spinors are vectors in ${\mathbb{C}^{4}}$, a complex vector space of dimension 4 that decomposes into two orthogonal 2-dimensional complex subspaces which are each invariant under the action of ${\textrm{Spin}(3,1)^{e}}$. Now, the even subalgebra ${C_{0}(3,1)\cong C(3,0)\cong\mathbb{C}(2)}$ can also be viewed as a complex vector space of dimension 4. The action of ${\textrm{Spin}(3,1)^{e}}$ on ${C_{0}(3,1)}$ by Clifford multiplication is linear, and ${C_{0}(3,1)}$ decomposes into two spaces invariant under ${\textrm{Spin}(3,1)^{e}}$: the bivectors that are real linear combinations of ${e_{0}e_{i}\cong\sigma_{i}}$ have negative determinant, while linear combinations of the remaining bivectors ${e_{i}e_{j}\cong\sigma_{i}\sigma_{j}=i\sigma_{k}}$ have positive determinant, as do the scalars. The pseudo-scalars are real multiples of ${\Omega=e_{0}e_{1}e_{2}e_{3}}$, and so have negative determinant. Thus an element of ${\textrm{Spin}^{e}(3,1)\cong SL(2,\mathbb{C})}$, having determinant ${+1}$, leaves invariant these positive and negative determinant subspaces under Clifford multiplication. Note that as ${C(3,0)}$, these subspaces are exactly the even and odd subspaces ${C_{0}(3,0)}$ and ${C_{1}(3,0)}$.

This alternative definition of spinor then readily generalizes to any dimension, i.e. spinors can be defined as elements of the ${2^{n-1}}$-dimensional vector space ${C_{0}(r,s)\cong C(r,s-1)}$ acted on by ${\textrm{Spin}(r,s)^{e}}$ via Clifford multiplication. However, it is important to note that despite the accidental equivalency in signature ${(3,1)}$, for other signatures and dimensions these definitions are quite distinct. In particular, the above decomposition of ${C_{0}(3,1)\cong\mathbb{C}(2)}$ as a vector space of spinors under the action of ${\textrm{Spin}(3,1)^{e}}$ is completely unrelated to the chiral decomposition of the Dirac rep ${C\mathbb{^{C}}(3,1)\cong\mathbb{C}(4)}$ when restricted to ${C_{0}(3,1)}$. This is underscored by the fact that while the Dirac rep decomposes as a rep of any part of the even subalgebra, the decomposition of the spinor space ${C_{0}(3,1)}$ only occurs under the action of the identity component ${\textrm{Spin}(3,1)^{e}}$: the determinant is not preserved under the action of ${\textrm{Spin}(3,1)\cong SL^{\pm}(2,\mathbb{C})}$, and so there is no decomposition in this case. Lastly, note that this gives us a new characterization of ${\textrm{Spin}\left(3,1\right)^{e}}$ as plus or minus the exponentials of the Lie algebra of vectors and bivectors in ${C(3,0)}$.