The differential geometry software package in Maple has the necessary tools and commands to automate the classification process for complex simple Lie algebras. The purpose of this thesis is to write the programs to complete the classification for real simple Lie algebras. This classification is difficult because the Cartan subalgebras are not all conjugate as they are in the complex case. For the process of the real classification, one must first identify a maximally noncompact Cartan subalgebra. The process of the Cayley transform is used to find this specific Cartan subalgebra. This Cartan subalgebra is used to find the simple roots for the given real simple Lie algebra. With this information, we can then create a Satake diagram. Then we match our given algebra's Satake diagram to a Satake diagram of a known algebra. The programs explained in this thesis complete this process of classification.

In this thesis we address the analytic and numerical aspects of isospectral flows. Such flows occur in mathematical physics and numerical linear algebra. Their main structural feature is to retain the eigenvalues in the solution space. We explore the solution of Isospectral flows and their stochastic counterpart using explicit generalisation of Magnus expansion. \par In the first part of the thesis we expand the solution of Bloch--Iserles equations, the matrix ordinary differential system of the form $ X'=[N,X^{2}],\ \ t\geq0, \ \ X(0)=X_0\in \textrm{Sym}(n),\ N\in \mathfrak{so}(n), $ where $\textrm{Sym}(n)$ denotes the space of real $n\times n$ symmetric matrices and $\mathfrak{so}(n)$ denotes the Lie algebra of real $n\times n$ skew-symmetric matrices. This system is endowed with Poisson structure and is integrable. Various important properties of the flow are discussed. The flow is solved using explicit Magnus expansion and the terms of expansion are represented as binary rooted trees deducing an explicit formalism to construct the trees recursively. Unlike classical numerical methods, e.g.\ Runge--Kutta and multistep methods, Magnus expansion respects the isospectrality of the system, and the shorthand of binary rooted trees reduces the computational cost of the exponentially growing terms. The desired structure of the solution (also with large time steps) has been displayed. \par Having seen the promising results in the first part of the thesis, the technique has been extended to the generalised double bracket flow $ X^{'}=[[N,X]+M,X], \ \ t\geq0, \ \ X(0)=X_0\in \textrm{Sym}(n),$ where $N\in \textrm{diag}(n)$ and $M\in \mathfrak{so}(n)$, which is also a form of an Isospectral flow. In the second part of the thesis we define the generalised double bracket flow and discuss its dynamics. It is noted that $N=0$ reduces it to an integrable flow, while for $M=0$ it results in a gradient flow. We analyse the flow for various non-zero values of $N$ and $M$ by assigning different weights and observe Hopf bifurcation in the system. The discretisation is done using Magnus series and the expansion terms have been portrayed using binary rooted trees. Although this matrix system appears more complex and leads to the tri-colour leaves; it has been possible to formulate the explicit recursive rule. The desired structure of the solution is obtained that leaves the eigenvalues invariant in the solution space.