Let \(B\) be a \(d\)-dimensional Brownian motion, with \(d \geq 2\) and \(\beta\) defined as

\[d \beta_{t}=\frac{1}{\left\|B_{t}\right\|} B_{t} \cdot d B_{t}=\frac{1}{\left\|B_{t}\right\|} \sum_{i=1}^{d} B_{t}^{i} d B_{t}^{i}, \quad \beta_{0}=0\]

Prove that \(\beta\) is a Brownian motion. This will be the starting point of the study of Bessel processes.