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A spinor is an element of a representation space for a Spin group; algebraically, spinor spaces arise from irreducible representations of a Clifford algebra or its even part. The Spin group is a double cover of a rotation group, so its action can distinguish rotations that ordinary vectors cannot.

For rotations in three-dimensional Euclidean space, a spinor can be a column in ${\mathbb{C}}^2$, acted on by $\mathrm{SU}(2)$. A rotation through angle $\theta$ about a unit direction $n$ acts as $\exp (-i\theta n\cdot \sigma /2)$, where $\sigma$ are the Pauli matrices. At $\theta =2\pi$ this is $-I$, and at $4\pi$ it is $I$. The sign change explains the distinction between spinors and spatial vectors and underlies the representation of spin one-half. A spinor is not a literal square root of a vector; the relationship is through the Clifford algebra action.

An irreducible complex Clifford algebra module in even spacetime dimension $2k$ has dimension $2^k$ and splits under the connected Spin group into two Weyl spinor modules of dimension $2^{k-1}$. In odd dimension $2k+1$, an irreducible spinor still has dimension $2^k$ but has no Weyl splitting. Reality conditions depend on signature and are additional to these complex counts.

For a Dirac spinor a spatial rotation acts in each chiral block by a matrix exponential $e^{\pm i\theta \mathbf{n}\cdot \mathbf{\sigma }/2}$, depending on angle orientation. Since $(\mathbf{n}\cdot \mathbf{\sigma }{)}^2=I_2$ for a unit axis, the Pauli matrix multiplication law gives this exponential as $\cos (\theta /2)I_2\pm i\sin (\theta /2)\mathbf{n}\cdot \mathbf{\sigma }$. At $\theta =2\pi$ it is $-I_2$, and at $4\pi$ it is $I_2$. A vector returns already at $2\pi$. This describes the lift of the rotation loop to the Spin group, rather than a single-valued representation of the ordinary rotation group on spinors.

For the B2 root system with long simple root first, the four-dimensional spinor representation has highest weight $(0,1)$. Its symmetric square has highest weight $(0,2)$ and dimension ten, so it is the Adjoint representation. The exterior square has the four nonzero defining weights and two zero weights; it splits into the five-dimensional vector representation of highest weight $(1,0)$ and a trivial representation. Thus the tensor square decomposes as displayed. The invariant alternating bilinear form reflects the pseudoreal representation structure. The spinor representation is a group representation of the Spin group $\mathrm{Spin}(5)$, while its tensor square descends to $SO(5)$.

A spinor representation is a representation of a Spin group, or infinitesimally of a Special orthogonal Lie algebra, constructed from a Clifford algebra. In even dimensions it can split into two chiral spinor representations.