We classify quadratic ${SL}_2\left(𝕂\right)$- and ${\mathrm{𝔰𝔩}}_2\left(𝕂\right)$-modules by crude computation, generalising in the first case a Theorem proved independently by F.G. Timmesfeld and S. Smith. The paper is the first of a series dealing with linearisation results for abstract modules of algebraic groups and associated Lie rings.