= Solution
For a real boundary point $x\ne0$, set
$$
X_t=\frac{g_t(x)-U_t}{\sqrt\kappa}.
$$
After a deterministic rescaling of time, the <Boundary-point Bessel flow for SLE> says that $X$ is a <Bessel process> of dimension
$$
\delta=1+\frac4\kappa.
$$
When $0<\kappa\leq4$, one has $\delta\geq2$, and the <Hitting-zero classification for a Bessel process> says that $X$ never reaches zero. Thus no nonzero real boundary point is swallowed. The standard Loewner trace criterion then implies that each new tip is attached only to the preceding tip and the trace never intersects its past, so it is simple. For $\kappa=0$, the equation is driven by zero and generates a vertical slit. Hence \b[$\operatorname{SLE}_\kappa$ is simple for $0\leq\kappa\leq4$].
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