  The aim of this note is to study \v{C}eby\v{s}\"ev JB$^*$-subtriples of
general JB$^*$-triples. It is established that if $F$ is a non-zero
\v{C}eby\v{s}\"ev JB$^*$-subtriple of a JB$^*$-triple $E$, then exactly one of
the following statements holds:\begin{enumerate}\item $F$ is a rank one
JBW$^*$-triple with dim$(F)\geq 2$ (i.e. a complex Hilbert space regarded as a
type 1 Cartan factor). Moreover, $F$ may be a closed subspace of arbitrary
dimension and $E$ may have arbitrary rank, \item $F= \mathbb{C} e$, where $e$
is a complete tripotent in $E$, \item $E$ and $F$ are rank two JBW$^*$-triples,
but $F$ may have arbitrary dimension, \item $F$ has rank greater or equal than
three and $E=F$. \end{enumerate}
