In order to apply to the non-associative structures the fundamental ideas of Sophus Lie, namely to assign to any local Lie group \(G\) its tangent object in the identity element -- its Lie algebra -- which determines \(G\) in a unique way, the author studies here a very wide class of geodesic loops with respect to a linear connection the curvature of which is zero [for the definition of this class see \textit{M. Kikkawa}, Hiroshima Math. J. 5, 141-179 (1955; Zbl 0304.53037), p. 160, and \textit{P. O. Mikheev} and \textit{L. V. Sabinin}, Quasigroups and Differential Geometry, Chapter XII in ``Quasigroups and Loops: Theory and Applications'', Sigma Ser. Pure Math. 8, 357-430 (1990; Zbl 0721.53018), p. 369]. The new tangential object used for this aim is called \(\Lambda\)-algebra: An \(n\)-dimensional real or complex vector space \(V\) is called \(\Lambda\)-algebra if there exists for every natural number \(p\) an algebraic map \(\Lambda_{(p,1)}:V\times V\to V\) such that the following properties are satisfied: 1. \(\Lambda_{(p,1)}(x,\lambda y)=\lambda \Lambda_{(p,1)} (x,y)\) for all \(\lambda \in K\) and \(x,y\in V\). 2. \(\Lambda_{(p,1)} (\lambda x,y)= \lambda^p\Lambda_{(p,1)}(x,y)\) for all \(\lambda \in K\) and \(x,y\in V\). 3. \(\sum^\infty_{p=1} \Lambda_{(p,1)}(x,y)\) converges on a neighbourhood \(N\) of 0 in \(V\). Specifically, the author proves that any geodesic loop \((L,\nabla,e)\) defines in the tangential space \(T_eL\) a unique \(\Lambda\)-algebra, and, moreover, that to any finite dimensional real \(\Lambda\)-algebra \(F\) there exists a geodesic loop \(L\), the \(\Lambda\)-algebra of which is isomorphic to \(F\). If the local loop \(L\) is diassociative, that is if every two elements generate a subgroup of \(L\), then to the \(\Lambda\)-algebra of \(L\) there corresponds a subseries of the Hausdorff-Campbell formula with respect to the binary Lie algebra belonging to \(L\).