  Lagrangian multiforms provide a variational framework to describe integrable
hierarchies. The case of Lagrangian $1$-forms covers finite-dimensional
integrable systems. We use the theory of Lie dialgebras introduced by
Semenov-Tian-Shansky to construct a Lagrangian $1$-form. Given a Lie dialgebra
associated with a Lie algebra $\mathfrak{g}$ and a collection $H_k$,
$k=1,\dots,N$, of invariant functions on $\mathfrak{g}^*$, we give a formula
for a Lagrangian multiform describing the commuting flows for $H_k$ on a
coadjoint orbit in $\mathfrak{g}^*$. We show that the Euler-Lagrange equations
for our multiform produce the set of compatible equations in Lax form
associated with the underlying $r$-matrix of the Lie dialgebra. We establish a
structural result which relates the closure relation for our multiform to the
Poisson involutivity of the Hamiltonians $H_k$ and the so-called ``double
zero'' on the Euler-Lagrange equations. The construction is extended to a
general coadjoint orbit by using reduction from the free motion of the
cotangent bundle of a Lie group. We illustrate the dialgebra construction of a
Lagrangian multiform with the open Toda chain and the rational Gaudin model.
The open Toda chain is built using two different Lie dialgebra structures on
$\mathfrak{sl}(N+1)$. The first one possesses a non-skew-symmetric $r$-matrix
and falls within the Adler-Kostant-Symes scheme. The second one possesses a
skew-symmetric $r$-matrix. In both cases, the connection with the well-known
descriptions of the chain in Flaschka and canonical coordinates is provided.
