Is the set of real matrices with at least one real logarithm closed under multiplication?

Question

Let $S$ be the set of real matrices with at least one real logarithm. For some couple of its elements, for example those with at least (one real logarithm each with submultiplicative norm smaller than $\ln 2/4$), the product of the couple can be proven to lie in $S$ via the Baker-Campbell-Hausdorff formula.

My question is more general- is S closed under multiplication as a whole? My guess is that one can use Lindelöf summation, which would converge on the Mittag-Leffler star(in this case, $S$ itself), on the BCH series, but I'm not sure if it works on matrix functions as well.

---

Simple proof: Apply Sylvester's formula to each term of the Lindelöf limit of the BCS series and pull the limit out(legal since Sylvester's formula is a finite series).

Answer 1

This is already not true for $2$-by-$2$ matrices: Consider $$ A = \begin{pmatrix}2 & 0 \\0 &\frac12\end{pmatrix}\quad \text{and}\quad B = \begin{pmatrix}-1 & 0 \\0 &-1\end{pmatrix}. $$ $A = \exp\bigl(\ln(2) K\bigr)$ and $B = \exp\bigl(\pi J\bigr)$ for $$ K= \begin{pmatrix}1 & 0 \\0 &-1\end{pmatrix}\quad\text{and}\quad J = \begin{pmatrix}0 & 1 \\-1 &0\end{pmatrix}. $$ However, $AB$ is not $\exp(X)$ for any real $2$-by-$2$ matrix $X$ because the trace of $AB$ is less than $-2$.