Example. A Euclidean unit normal [lawson2016spin, I.2, (2.12), p. 14; (2.24)--(2.26), p. 18] [fcap-000D]
Example. A Euclidean unit normal [lawson2016spin, I.2, (2.12), p. 14; (2.24)--(2.26), p. 18] [fcap-000D]
Let \(V=\mathbb {R}^n\) with Euclidean inner product and use the sign convention \[Q(x)=-\langle x,x\rangle .\] For a unit vector \(v\), one has \(Q(v)=-1\); thus \(c=1\) satisfies the square condition in Lemma [fcap-000B]. The Clifford generator \(\iota (v)\) is already Pin-normalized, and its action is the familiar hyperplane reflection \[\rho _v(w)=w-2\langle v,w\rangle v.\] For orthonormal \(v,w\), the even product \(\iota (v)\iota (w)\) lies in Spin and lifts the composition \(\rho _v\rho _w\). This is the Euclidean specialization of the general Pin and paired-Spin lifting statements in Lemma [fcap-000B] and Theorem [fcap-000C].