Which dimension is needed to represent a Lie algebra as a matrix algebra (as in Ado's Thm)? Suppose $\mathfrak g$ is a real finite dimensional Lie algebra. If I understand it correctly, Ado's Theorem states that there is a real vector space $V$ and an injective Lie algebra homomorphism $\pi:\mathfrak g \to \mathrm{End}(V)$.
My question is what the smallest dimension of the vector space $V$ is that is needed. I believe that if $\mathfrak g$ has only a trivial center, that then the answer is $\mathrm{dim}(\mathfrak g)$, because one can use the adjoint representation. But is this really the minimal one?
 A: No, this need not be the minimal one.
Let $\mathfrak{g}$ be an $n$-dimensional Lie algebra over a field $K$ of characteristic zero. Define an invariant of $\mathfrak{g}$ by
$$\mu (\mathfrak{g}):=\min \{\dim_K (M) \mid M \text{ is a faithful $\mathfrak{g}$-module}\}.$$
For complex simple Lie algebras $\mathfrak{s}$, for example, we always have $\mu (\mathfrak{s})< \dim (\mathfrak{s})$, except for type $E_8$. Here is a small table over $\Bbb C$.
\begin{array}{c|c|c}
\mathfrak{g} & \dim (\mathfrak{g}) & \mu(\mathfrak{g}) \\
\hline
A_n, \; n\ge 1 & (n+1)^2-1 & n+1 \\
B_2 & 10 & 4 \\
B_n, \; n\ge 3 & 2n^2+n & 2n+1 \\
C_n, \; n\ge 3 & 2n^2+n & 2n \\
D_n, \; n\ge 4 & 2n^2-n & 2n \\
E_6 & 78 & 27 \\
E_7 & 133 & 56 \\
E_8 & 248 & 248 \\
F_4 & 52 & 26 \\
G_2 & 14 & 7 \\
\end{array}
In general, the question about the minimal dimension is quite difficult and we only have lower and upper bounds. For references see the papers (my papers and other papers) in this article.
For example, let $\mathfrak{g}$ be a nilpotent Lie algebra of dimension $n$ and nilpotency class
$k$. Denote by $p(j)$ the number of partitions of $j$ and let
$$p(n,k)=\sum_{j=0}^{k}\binom{n-j}{k-j}p(j).$$
Then $\mu(\mathfrak{g})\le p(n,k)$. If $k=n-1$, with $n\ge 3$, then $n\le \mu(\mathfrak{g})$.
