  In the present paper we set up a connection between the indices of the Tits
algebras of a simple linear algebraic group $G$ and the degree one parameters
of its motivic $J$-invariant. Our main technical tool are the second Chern
class map and Grothendieck's $\gamma$-filtration.
  As an application we recover some known results on the $J$-invariant of
quadratic forms of small dimension; we describe all possible values of the
$J$-invariant of an algebra with orthogonal involution up to degree 8 and give
explicit examples; we establish several relations between the $J$-invariant of
an algebra $A$ with orthogonal involution and the $J$-invariant of the
corresponding quadratic form over the function field of the Severi-Brauer
variety of $A$.
