I will consider products of arbitrary i.i.d. random real $2 times 2$ matrices that are close to the identity matrix. I will first identify a physical model of one-dimensional disordered quantum mechanics related to the most general matrix products. Using the Iwasawa decomposition of $SL(2,R)$, we can identify a continuum regime where the mean values and the covariances of the three Iwasawa parameters are simultaneously small. In this regime, the Lyapunov exponent of the product may be systematically obtained and is shown to present a scaling form. This general analysis thus allows us, among other things, to recover in a unified framework many results known previously from exactly solvable models of one-dimensional disordered systems and find several new ones. \ References : \ - Alain Comtet, Christophe Texier and Yves Tourigny, Products of random matrices and generalised quantum point scatterers, J. Stat. Phys. {bf 140} (3), 427--466 (2010). \ - Alain Comtet, Jean-Marc Luck, Christophe Texier and Yves Tourigny, The Lyapunov exponent of products of random $2times2$ matrices close to the identity, J. Stat. Phys. {bf 150} (1), 13--65 (2013).