On a Theorem of Kadets and Pełczyński

Necessary and sufficient conditions are found under which a symmetric space $X$ on $[0,1]$ of type $2$ has the following property, which was first proved for the spaces $L_p$, $p>2$, by Kadets and Pełcziński: if $\{u_n\}_{n=1}^\infty$ is an unconditional basic sequence in $X$ such that
$$ \|u_n\|_X\asymp\|u_n\|_{L_1},\qquad n\in\mathbb N, $$
then the norms of the spaces $X$ and $L_1$ are equivalent on the closed linear span $[u_n]$ in $X$. For sequences of martingale differences, this implication holds in any symmetric space of type $2$.