Vijay-Shanker and Weir have shown in \cite{VSW94} that Tree Adjoining Grammars and Combinatory Categorial Grammars can be transformed into equivalent Linear Indexed Grammars (LIGs) which can be recognized in ${\cal O}(n^6)$ time using a Cocke-Kasami-Younger style algorithm. This paper exhibits another recognition algorithm for LIGs, with the same upper-bound complexity, but whose average case behaves much better. This algorithm works in two steps: first a general context-free parsing algorithm (using the underlying context-free grammar) builds a shared parse forest, and second, the LIG properties are checked on this forest. This check is based upon the composition of simple relations and does not require any computation of symbol stacks.