In this article, we show the existence of large sets LS2[3](2,k,v) for infinitely many values of k and v. The exact condition is v≥8 and 0≤k≤v such that for the remainders v¯ and k¯ of v and k modulo 6 we have 2≤v¯. The proof is constructive and consists of two parts. First, we give a computer construction for an LS2[3](2,4,8), which is a partition of the set of all 4-dimensional subspaces of an 8-dimensional vector space over the binary field into three disjoint 2-(8,4,217)2 subspace designs. Together with the already known LS2[3](2,3,8), the application of a recursion method based on a decomposition of the Graßmannian into joins yields a construction for the claimed large sets. [ABSTRACT FROM AUTHOR]