 Exponential-surjectivity obstruction from distinct negative eigenvalues

A real matrix with at least one simple negative eigenvalue cannot be a matrix exponential of a real matrix: a logarithm would commute with it and preserve each one-dimensional real negative-eigenvalue space, where exponentiation can only produce a positive eigenvalue. In particular $\mathrm{diag}(-2,-1/2)$ belongs to the connected special linear group ${\mathrm{SL}}_2(\mathbb{R})$ but has no real logarithm.