Computing representations on the enveloping algebra was developed by physicists (e.g. \textit{B. Gruber} and \textit{A. U. Klimyk} [J. Math. Phys. 25, 755-764 (1984; Zbl 0548.17005)]), but their methods work only for very simple Lie algebras. This new algorithm constructs explicit \textit{principal matrices} and avoids difficulties with commutators. The adjoint group [\textit{C. Chevalley}, Theory of Lie groups. I. Princeton Mathematical Series, Vol.~8. Princeton University Press. (1946; Zbl 0063.00842)] comes for free. Denote by \(\{\xi_1,\xi_2,\dots,\xi_d\}\) a basis for the given Lie algebra. A basis for its universal enveloping algebra is then \[ [[n]]\overset\text{def} = \xi^{n_1}_1\xi^{n_2}_2\cdots \xi^{n_d}_d, \] where \(n\equiv (n_1,\dots, n_d)\). Group elements in a neighbourhood of the identity can be expressed as \[ g(\alpha)\overset\text{def}= e^{\sum_\mu\alpha_\mu\xi_\mu}=g(A)\overset\text{def}= e^{A_1\xi_1}e^{A_2\xi_2}\dots e^{A_d\xi_d}. \] The \(\alpha_i\), \(A_i\) are called respectively \textit{coordinates of the first and second kind.} The \textit{dual representations} are vector fields in terms of the coordinates of the second kind acting on the left or right, i.e. they define the left \textit{principal matrix} \(\pi^{\ddagger}(A)\) by \[ \xi_jg(A)=\sum_\mu\pi^\ddagger_{j\mu}(A)\partial_\mu g(A) \] leading to the dual representation \[ \xi^\ddagger_j \overset\text{def}= \sum_\mu\pi^\ddagger_{j\mu}(A)\partial_\mu. \] If \(A\) depends on a parameter \(s\) then, for any function \(f(A)\), there is a flow \(\dot f =\sum_\mu\dot A_\mu\partial_\mu f\) where \(\dot{\;}\equiv d/ds\). Letting \(X = \sum_{\mu}\alpha_\mu\xi_\mu\) leads to \[ \dot g = Xg =\sum_{\mu}\alpha_\mu\xi^\ddagger_\mu g=\sum_{\lambda\mu}\alpha_\lambda \pi^\ddagger_{\lambda\mu} \partial_\mu g \] and similarly in terms of \(\pi^*\), \(\xi^*\) with \(X\) acting on the right. Matrix elements are defined by group elements \(g(A,\xi)\) acting on the basis \([[n]]\). Using the \(\pi\)-matrices, the adjoint group, which is the exponential of the adjoint representation, is readily computed as follows. Denote the transposes by \(\widehat\pi= (\pi^\ddagger)^t\), \(\widehat\pi^*=(\pi^*)^t\) and let \(\check\pi(A)\) denote the adjoint group \(g(A,\check\xi)\), which is \(g\) with each basis element \(\xi_i\) replaced by the corresponding matrix of the adjoint representation \(\check\xi_i\). Then the main theorem, proved in \textit{P. Feinsilver} and \textit{R. Schott} [Algebraic structures and operator calculus, Vol. 3: Representations of Lie groups. Kluwer Academic Publishers. (1996; Zbl 0885.22014)] is that the adjoint group \(q(A,\check\xi)\) satisfies \(\widehat\pi^* =\widehat{\pi}g(A,\check\xi)\), i.e. \(\check\pi=\widehat\pi^{-1}\widehat{\pi}^*\). The transposes \(\widehat{\pi}\), \(\widehat{\pi}^*\) are used in defining double duals in which the derivatives \(\partial_i\) are replaced by variables \(y_i\) and the variables \(A_i\) are replaced by derivatives \(\partial_i=\partial/\partial y_i\), thus \[ \widehat\xi_j\overset\text{def}=\sum_\mu y_\mu\widehat\pi_{\mu j}(\partial), \] where \(\partial = (\partial_1,\dots,\partial_d)\) and similarly for \(\xi^*\), \(\pi^*\). The goal is to compute the matrices \(\pi^\ddagger\), \(\pi^*\). The direct approach using commutation relations is generally quite difficult, whereas the following algorithm avoids these difficulties and the use of the adjoint action. 1. Given a basis \(\{\xi_i\}\) form \(X = \sum_i\alpha_i\xi_i\) (or given \(X\) compute \(\{\xi_i\}\) as \(\{\partial X/\partial\alpha_i\})\). 2. Compute \(g\) as the product \(e^{A_1\xi_1}e^{A_2\xi_2}\dots e^{A_d\xi_d}\). 3. Compute \(Xg\) by matrix multiplication. 4. Write \(\dot g\) formally and equate it to \(Xg\). 5. Solve for \(\dot A_i\) and form the row vector \(\dot A\). 6. Find \(\pi^\ddagger\) from \(\dot A = \alpha\pi^\ddagger\) with \(\alpha\) the row vector \((\alpha_1,\dots,\alpha_d)\). 7. Repeat steps 3-6 with \(gX\) to find \(\pi^*\). 8. Form the vector fields \(\xi^\ddagger=\pi^\ddagger\partial\), \(\xi^* =\pi^*\partial\) using column vectors. 9. Form double duals \(\widehat\xi\), \(\widehat\xi^*\) using \(\widehat\pi=(\pi^\ddagger)^t\), \(\widehat{\pi}^*=(\pi^*)^t\). 10. Compute the adjoint group \(\check \pi=\widehat\pi^{-1}\widehat\pi^*\). A detailed example is given for E2, the Euclidean group in two dimensions, and the left-dual is given for E3. This approach involves much smaller matrices than does the direct approach. A complete and straightforward implementation in Maple is given.