  We explore the optimality of the constants making valid the recently
established Little Grothendieck inequality for JB$^*$-triples and
JB$^*$-algebras. In our main result we prove that for each bounded linear
operator $T$ from a JB$^*$-algebra $B$ into a complex Hilbert space $H$ and
$\varepsilon>0$, there is a norm-one functional $\varphi\in B^*$ such that
$$\|Tx\|\le(\sqrt{2}+\varepsilon)\|T\|\|x\|_\varphi\quad\mbox{ for }x\in B.$$
The constant appearing in this theorem improves the best value known up to date
(even for C$^*$-algebras). We also present an easy example witnessing that the
constant cannot be strictly smaller than $\sqrt2$, hence our main theorem is
`asymptotically optimal'. For type I JBW$^*$-algebras we establish a canonical
decomposition of normal functionals which may be used to prove the main result
in this special case and also seems to be of an independent interest. As a tool
we prove a measurable version of the Schmidt representation of compact
operators on a Hilbert space.
